researchseminars.org

Yoshinori Gongyo (The University of Tokyo)
2020-04-23 16:00 - 17:00 UTC | https://researchseminars.org/talk/ZAG/1/
Title: On a generalized Batyrev's cone conjecture (https://researchseminars.org/talk/ZAG/1/)
by Yoshinori Gongyo (The University of Tokyo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract: TBA

Christopher Hacon (The University of Utah)
2020-04-28 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/2/
Title: Recent progress in the MMP for 3-folds and 4-folds in char p>0 (https://researchseminars.org/talk/ZAG/2/)
by Christopher Hacon (The University of Utah) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract: TBA

Arnaud Beauville (Université de Nice)
2020-04-30 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/3/
Title: Vector bundles on Fano threefolds and K3 surfaces (https://researchseminars.org/talk/ZAG/3/)
by Arnaud Beauville (Université de Nice) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract: TBA

Chen Jiang (Shanghai Center for Mathematical Sciences, Fudan University)
2020-05-05 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/4/
Title: Minimal log discrepancies of 3-dimensional non-canonical singularities (https://researchseminars.org/talk/ZAG/4/)
by Chen Jiang (Shanghai Center for Mathematical Sciences, Fudan University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Canonical and terminal singularities, introduced by Reid, appear naturally in minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension 3, while the property of non-canonical singularities is still mysterious. We investigate the difference between canonical and non-canonical singularities via minimal log discrepancies (MLD). We show that there is a gap between MLD of 3-dimensional non-canonical singularities and that of 3-dimensional canonical singularities, which is predicted by a conjecture of Shokurov. This result on local singularities has applications to global geometry of Calabi–Yau 3-folds. We show that the set of all non-canonical klt Calabi–Yau 3-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3-folds are bounded from above.

Samuel Grushevsky (Stony Brook University)
2020-05-07 18:00 - 19:00 | https://researchseminars.org/talk/ZAG/5/
Title: Geometry of moduli of cubic threefolds (https://researchseminars.org/talk/ZAG/5/)
by Samuel Grushevsky (Stony Brook University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The moduli space of cubic threefolds can be thought of as a GIT quotient of the projective space of all cubic polynomials, studied via the period map to a ball quotient, or via the intermediate Jacobians. We describe the relations between various compactifications of the moduli space of cubic threefolds that arise in these ways, and compute their cohomology. Based on joint works with S. Casalaina-Martin, K. Hulek, R. Laza.

Kristin De Vleming (University of California, San Diego)
2020-05-12 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/6/
Title: Wall crossing for K-moduli spaces of plane curves (https://researchseminars.org/talk/ZAG/6/)
by Kristin De Vleming (University of California, San Diego) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
This talk will focus on compactifications of the moduli space of smooth plane curves of degree d at least 4. We will regard a plane curve as a log Fano pair (P2, aC), where a is a rational number, and study the compactifications arising from K stability for these pairs and log Fano pairs in general. We establish a wall crossing framework to study these spaces as a varies and show that, when a is small, the moduli space coming from K stability is isomorphic to the GIT moduli space. We describe all wall crossings for degree 4, 5, and 6 plane curves and discuss the picture for general Q-Gorenstein smoothable log Fano pairs. This is joint work with Kenneth Ascher and Yuchen Liu.

John Christian Ottem (University of Oslo)
2020-05-14 15:30 - 16:30 | https://researchseminars.org/talk/ZAG/7/
Title: Tropical degenerations and stable rationality (https://researchseminars.org/talk/ZAG/7/)
by John Christian Ottem (University of Oslo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract: TBA

Mircea Mustață (University of Michigan)
2020-05-19 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/11/
Title: Minimal exponent and Hodge filtrations (https://researchseminars.org/talk/ZAG/11/)
by Mircea Mustață (University of Michigan) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will discuss an invariant of singularities, Saito's minimal exponent, and its connections with various other invariants of singularities. The minimal exponent is a refinement of the log canonical threshold that can be used to also measure rational hypersurface singularities. This is based on joint work with Mihnea Popa.

Zsolt Patakfalvi (École polytechnique fédérale de Lausanne)
2020-05-26 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/12/
Title: On the Beauville-Bogomolov decomposition in positive characteristic (https://researchseminars.org/talk/ZAG/12/)
by Zsolt Patakfalvi (École polytechnique fédérale de Lausanne) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Abstract: I will present a joint with Maciej Zdanowicz towards a positive characteristic version of the Beauville-Bogomolov decomposition. Over the complex numbers this decomposition was shown using differential geometry methods in the 70's and in the 80's. It concerns varieties with trivial canonical bundle, which we call K-trivial here. The main statement over the complex number is that smooth projective K-trivial varieties admit an etale cover which splits as a product of three types of varieties: abelian, Calabi-Yau and symplectic. I will present a similar statement in positive characteristic for (weakly) ordinary K-trivial varieties, the proof of which uses purely positive characteristic methods.

Xavier Roulleau (University of Aix-Marseille)
2020-05-21 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/13/
Title: On the geometric models of K3 surfaces with finite automorphism group and Picard number larger than two (https://researchseminars.org/talk/ZAG/13/)
by Xavier Roulleau (University of Aix-Marseille) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Vinberg and Nikulin classified K3 surfaces which have finite automorphism group and Picard number 4 and 3,5,..,19 respectively. That classification is lattice theoretic, according to the Neron-Severi group of these surfaces; there are 118 such lattices. In this talk I will discuss on the geometric construction of these surfaces (by double coverings or complete intersections) and describe their (finite) set of (-2)-curves, which gives the ample cone. Most of the moduli spaces of these K3 surfaces are unirational. A part of this talk is based on a joint work with Michela Artebani and Claudia Correa Diesler.

Alexandra Kuznetsova (Higher School of Economics)
2020-05-28 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/18/
Title: Sextic double solids (https://researchseminars.org/talk/ZAG/18/)
by Alexandra Kuznetsova (Higher School of Economics) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Abstract: One of the first examples of unirational non-rational threefold was provided by Artin and Mumford and it was a double cover of P^3 branched in a nodal quartic surface, so called quartic double solid.
Then Endrass studied this class of varieties and showed that the example by Artin and Mumford gives a unique family of non-rational nodal quartic double solids. I am going to tell about the next interesting class of threefolds --- nodal sextic double solids. I will describe 4 families of them such that any non-rational variety of this type lies in one of those families and explain the proof.

Stefan Schreieder (Leibniz University)
2020-06-02 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/19/
Title: Equality in the Bogomolov-Miyaoka-Yau inequality in the non-general type case (https://researchseminars.org/talk/ZAG/19/)
by Stefan Schreieder (Leibniz University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We classify all good minimal models of dimension n and with vanishing Chern number $c_1^{n-2}c_2(X)=0$, which corresponds to equality in the Bogomolov-Miyaoka—Yau inequality in the non-general type case. Here the most interesting case is that of Kodaira dimension n-1, where any minimal model is known to be good. Our result solves completely a problem a Kollar. In dimension three, our approach together with previous work of Grassi and Kollar also leads to a complete solution of a conjecture of Kollar, asserting that on a minimal threefold, c_1c_2 is either zero or universally bounded away from zero. Joint work with Feng Hao.

Caucher Birkar (University of Cambridge)
2020-06-04 13:00 - 14:00 | https://researchseminars.org/talk/ZAG/20/
Title: Geometry of polarised varieties (https://researchseminars.org/talk/ZAG/20/)
by Caucher Birkar (University of Cambridge) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will talk about projective varieties polarised by ample divisors (or more generally nef and big divisors) in particular from a birational geometry point of view, and present some recent results in this direction.

Yuri Prokhorov (Moscow State University)
2020-06-09 15:30 - 16:30 | https://researchseminars.org/talk/ZAG/21/
Title: General elephants for 3-fold extremal contractions (https://researchseminars.org/talk/ZAG/21/)
by Yuri Prokhorov (Moscow State University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will discuss effective results on the classification of extremal contractions in the 3-dimensional MMP. In particular, I will present some recent result based on joint work with Shigefumi Mori on the existence of general elephants.

Ilya Zharkov (Kansas State University)
2020-06-11 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/22/
Title: Topological SYZ fibrations with discriminant in codimension 2 (https://researchseminars.org/talk/ZAG/22/)
by Ilya Zharkov (Kansas State University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
To date only for K3 surfaces (trivial) and the quintic threefold (due to M. Gross) the discriminant can be made to be in codimension two. I will outline the source of the problem and how to resolve it in much more general situations using phase and over-tropical pairs-of-pants. Joint project with Helge Ruddat.

Nikolaos Tziolas (University of Cyprus)
2020-06-16 15:30 - 16:30 | https://researchseminars.org/talk/ZAG/23/
Title: Vector fields on canonically polarized surfaces (https://researchseminars.org/talk/ZAG/23/)
by Nikolaos Tziolas (University of Cyprus) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk I will present some results about the geometry of canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, equivalently non reduced automorphism scheme, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces.

Claire Voisin (Collège de France)
2020-06-18 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/24/
Title: Triangle varieties and surface decomposition of hyper-Kahler manifolds (https://researchseminars.org/talk/ZAG/24/)
by Claire Voisin (Collège de France) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In recent years, new constructions of complete families of polarized hyper-Kahler manifolds have been found starting from Fano geometry. These hyper-Kahler manifolds also appear as general deformations of Hilbert schemes of K3 surfaces or O'Grady manifolds. I will introduce the notion of surface decomposition for a variety X with a nontrivial Hodge structure on degree 2 cohomology. I will show that this notion is restrictive topologically, as it implies Beauville-Fujiki type relations. I will also show the existence of such a surface decomposition for the general hyper-Kahler manifolds mentioned above. This has interesting consequences on Beauville's conjecture on the Chow ring of hyper-Kahler manifolds.

Alexander Polishchuk (University of Oregon)
2020-06-23 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/25/
Title: Hyperelliptic limits of quadrics through canonical curves and the super-Schottky locus (https://researchseminars.org/talk/ZAG/25/)
by Alexander Polishchuk (University of Oregon) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will describe joint works with Eric Rains and with Giovanni Felder and David Kazhdan. The first part will be about a classical topic of quadrics through canonically embedded curves. We study limiting quadrics as canonical curves approach a hyperelliptic limit. There is a surprizingly simple description of all such limits. I will also discuss the connection to ribbon curves (which are thickenings of rational normal curves) and to the blow up of the moduli space of curves at the hyperelliptic locus. In the second part I will talk about the super-period map for supercurves and the calculation of its infinitesimal variation. This variation is given by a natural Massey product that can be defined for any curve with a theta-characteristic. Combining this with the result of part 1 we get some information about the super-Schottky locus.

David Mumford (Harvard University and Brown University)
2020-06-25 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/26/
Title: A moduli space in the differential geometry world (https://researchseminars.org/talk/ZAG/26/)
by David Mumford (Harvard University and Brown University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The space of simple closed smooth plane curves is an infinite dimensional manifold and supports a great diversity of Riemannian metrics. They have very diverse curvature properties and even include universal Teichmuller space. I want to talk in particular about a recent example: modeling 2D waves in water (aka gravity waves) that some believe explains so-called rogue waves.
After the talk, we plan to have Q&A session at 16:00 GMT. If you have a question for Prof. Mumford, let Ivan Cheltsov know in advance (by e-mail).

Izzet Coskun (University of Illinois at Chicago)
2020-06-30 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/27/
Title: The stabilization of the cohomology of moduli spaces of sheaves on surfaces (https://researchseminars.org/talk/ZAG/27/)
by Izzet Coskun (University of Illinois at Chicago) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The Betti numbers of the Hilbert scheme of points on a smooth, irreducible projective surface have been computed by Gottsche. These numbers stabilize as the number of points tends to infinity. In contrast, the Betti numbers of moduli spaces of semistable sheaves on a surface are not known in general. In joint work with Matthew Woolf, we conjecture these also stabilize and that the stable numbers do not depend on the rank. We verify the conjecture for large classes of surfaces. I will discuss our conjecture and provide the evidence for it.

Susanna Zimmermann (Université Angers)
2020-07-02 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/28/
Title: Finite quotients of Cremona groups (https://researchseminars.org/talk/ZAG/28/)
by Susanna Zimmermann (Université Angers) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The Cremona group is the group of birational self-maps of the projective space, and it is very very big. While in dimension 2 over algebraically closed fields there are no finite quotients of this group, there are many such quotients over non-closed fields and in higher dimension. I will discuss why this is and how these quotients come up.

Ziquan Zhuang (MIT)
2020-07-07 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/29/
Title: K-stability of Fano varieties via admissible flags (https://researchseminars.org/talk/ZAG/29/)
by Ziquan Zhuang (MIT) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I'll present a general approach to prove the K-stability of explicit Fano varieties. Among the applications, we confirm the existence of K\"ahler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, calculate the stability thresholds of some Fano varieties and provide a counterexample to the Higher Rank Finite Generation conjecture. Based on joint work with Hamid Ahmadinezhad.

Harold Blum (University of Utah)
2020-07-09 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/30/
Title: On properness of K-moduli spaces and destabilizations of Fano varieties (https://researchseminars.org/talk/ZAG/30/)
by Harold Blum (University of Utah) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
K-stability is an algebraic notion that detects when a smooth Fano variety admits a Kahler-Einstein metric. Recently, there has been significant progress on constructing moduli spaces of K-polystable Fano varieties using algebraic methods. One of the remaining open problems is to show that these moduli spaces are proper. In this talk, I will discuss work with Daniel Halpern-Leistner, Yuchen Liu, and Chenyang Xu, in which we reduce the properness of such K-moduli spaces to the existence of certain optimal destabilization of Fano varieties.

Hélène Esnault (Freie Universität Berlin)
2020-07-14 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/31/
Title: Density of arithmetic representations (https://researchseminars.org/talk/ZAG/31/)
by Hélène Esnault (Freie Universität Berlin) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The lecture surveys recent work with Moritz Kerz. The motivation is the conjecture that the Hard-Lefschetz (HL) property holds on smooth projective varieties defined over algebraically closed char. $p>0$ fields for cohomology with values in semi-simple $\ell$-adic local systems $V$. We know it is true if $V$ comes from geometry (Deligne, Beilinson-Bernstein-Deligne-Gabber) by Deligne’s theory of weights. In absence of weights, we proved it if $V$ has rank $1$ and reduced the whole HL conjecture to a density conjecture on arithmetic semi-simple $\ell$-adic systems on $P^1$ minus $3$ closed points, which we can prove in rank $2$.

Matthias Schuett (Leibniz Universität Hannover)
2020-07-16 15:30 - 16:30 | https://researchseminars.org/talk/ZAG/32/
Title: Rational curves on Enriques surfaces, but only few (https://researchseminars.org/talk/ZAG/32/)
by Matthias Schuett (Leibniz Universität Hannover) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Rational curves play a fundamental role for the structure of an Enriques surface. I will first review the general theory before focussing on the case of low degree rational curves. To this end, I will discuss joint work with S. Rams (Krakow) which develops an explicit sharp bound on the number of rational curves of given degree relative to the degree of the surface. The proof builds on a general argument in parallel to the case of K3 surfaces which allows us to extend bounds of Miyaoka and Degtyarev.

Jihun Park (POSTECH)
2020-07-21 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/33/
Title: Cayley octads, plane quartic curves, Del Pezzo surfaces of degree 2 and double Veronese cones (https://researchseminars.org/talk/ZAG/33/)
by Jihun Park (POSTECH) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
A net of quadrics in the 3-dimensional projective space whose singular members are parametrized by a smooth plane quartic curve has exactly eight distinct base points, called a regular Cayley octad. It is a classical result that there is a one-to-one correspondence between isomorphism classes of regular Cayley octads and isomorphism classes of smooth plane quartic curves equipped with even theta-characteristics. We can also easily observe a one-to-one correspondence between isomorphism classes of smooth plane quartic curves and isomorphism classes of smooth Del Pezzo surfaces of degree 2. In this talk, we set up a one-to-one correspondence between isomorphism classes of smooth plane quartic curves and isomorphism classes of double Veronese cones with 28-singular points. Also, we explain how the 36 even theta characteristics of a given smooth quartic curve appear in the corresponding double Veronese cone. This is a joint work with Hamid Ahmadinezhad, Ivan Cheltsov and Constantin Shramov.

Ruadhai Dervan (University of Cambridge)
2020-07-23 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/34/
Title: Stability of fibrations (https://researchseminars.org/talk/ZAG/34/)
by Ruadhai Dervan (University of Cambridge) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The notion of K-stability of a polarised variety has been heavily studied in recent years, due to its link both with moduli theory (one should be able to form moduli spaces of K-stable varieties) and to Kahler geometry (K-stability should be equivalent to the existence of a constant scalar curvature Kahler metric on the variety). This story has been particularly successful for Fano varieties. I will describe a notion of stability for polarised fibrations, which generalises K-stability of polarised varieties when the base of the fibration is a point, and slope stability of a vector bundle when the variety is the projectivisation of a vector bundle. I will speculate that one should be able to form moduli spaces of stable fibrations, much as one can form moduli spaces of slope stable vector bundles over a fixed base. The main result, however, will be a description of the link with certain canonical metrics on fibrations. This is joint work with Lars Sektnan.

Jean-Pierre Demailly (Université Grenoble Alpes)
2020-07-28 15:30 - 16:30 | https://researchseminars.org/talk/ZAG/35/
Title: Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles (https://researchseminars.org/talk/ZAG/35/)
by Jean-Pierre Demailly (Université Grenoble Alpes) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths - and even in the stronger dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled. We also discuss a new concept of volume for vector bundles.

Ziwen Zhu (University of Utah)
2020-07-30 15:00 - 16:30 | https://researchseminars.org/talk/ZAG/36/
Title: Equivariant K-stability under finite group action (https://researchseminars.org/talk/ZAG/36/)
by Ziwen Zhu (University of Utah) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Equivariant K-stability is defined via equivariant test configurations. By definition it is weaker than the usual K-stability and for varieties with large symmetry, it is often easier to check equivariant K-stability. For reductive group action, it is conjectured that equivariant K-polystability implies K-polystability. In this talk, I will discuss recent results about equivariant K-stability and present a proof of the conjecture for finite group action. The talk is based on joint work with Yuchen Liu.

Hamid Ahmadinezhad (Loughborough University)
2020-08-04 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/37/
Title: Birational geometry of Fano 3-fold hypersurfaces of higher index (https://researchseminars.org/talk/ZAG/37/)
by Hamid Ahmadinezhad (Loughborough University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will speak about an approach to birational classification of Fano 3-folds, post MMP. As a part of this general guideline, I will highlight some recent results about birational geometry of Fano hypersurfaces of higher index. The latter is a joint work with Ivan Cheltsov and Jihun Park.

Marina Logares (Universidad Complutense de Madrid)
2020-08-06 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/38/
Title: Poisson and symplectic geometry of the moduli spaces of Higgs bundles (https://researchseminars.org/talk/ZAG/38/)
by Marina Logares (Universidad Complutense de Madrid) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will talk about some natural Poisson and symplectic properties of the moduli spaces of Higgs bundles when some extra structure, such as a framing, is added. This is an overview of various past and ongoing work with I. Biswas, J. Martens, A. Peón-Nieto and S. Szabó. I will not assume any previous knowledge on the subject.

Robert Lazarsfeld (Stony Brook University)
2020-08-11 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/39/
Title: Cayley-Bacharach theorems and multiplier ideals (https://researchseminars.org/talk/ZAG/39/)
by Robert Lazarsfeld (Stony Brook University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Cayley-Bacharach theorems originate in the classical statement if two plane curves of degrees c and d meet in cd points, then any curve of degree (c + d - 3) passing through all but one of these points must also pass through the remaining one. Following work of Griffiths and Harris in the 1970s, one now sees this as a special case of a general result about zero-loci of sections of a vector bundle. I will explain how bringing multiplier ideals into the picture leads (for free) to a variant that allows for excess vanishing. This is joint work with Lawrence Ein.

Yuchen Liu (Yale University)
2020-08-13 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/40/
Title: On K-stability of cubic hypersurfaces (https://researchseminars.org/talk/ZAG/40/)
by Yuchen Liu (Yale University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
K-stability of Fano varieties is an algebro-geometric stability condition characterizing the existence of K\"ahler-Einstein metrics. Recent progress on K-stability suggests that it provides a good moduli theory for Fano varieties. In this talk, I will explain how K-moduli spaces can help us prove K-stability of smooth cubic hypersurfaces in dimension at most 4, using a local-to-global volume comparison result. Part of this talk is based on joint work with Chenyang Xu.

Alessio Corti (Imperial College London)
2020-08-18 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/41/
Title: Smoothing Gorenstein toric affine 3-folds (https://researchseminars.org/talk/ZAG/41/)
by Alessio Corti (Imperial College London) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will state a conjecture on the smoothing components of the deformation space, and discuss one or more of the following topics: possible strategies for proving it, applications to the Fanosearch program, global and higher dimensional analogs. The talk is based on a recent collaboration with Andrea Petracci and Matej Filip.

Nicola Pagani (University of Liverpool)
2020-08-20 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/42/
Title: Classifying fine compactified universal Jacobians (https://researchseminars.org/talk/ZAG/42/)
by Nicola Pagani (University of Liverpool) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We introduce the notion of a fine compactified Jacobian of a nodal curve, as an arbitrary compact open subspace of the moduli space of rank-1 torsion-free simple sheaves. We show that fine compactified Jacobians correspond to a certain combinatorial datum, which is obtained by only keeping track, for all sheaves, of (1) the locus where it fails to be locally free, and (2) its multidegree. This notion generalizes to flat families of curves, and so does its combinatorial counterpart. When the family is the universal family over the moduli space of curves, we have the following results: (a) in the absence of marked points, we can fully classify these combinatorial data and deduce that the only fine compactified universal Jacobians are the classical ones (which were constructed by Pandharipande and Simpson in the nineties) and (b) in the presence of marked points there are exotic (and new) examples that cannot be obtained as compactified universal Jacobians associated to a polarization. This is a joint work in progress with Jesse Kass.

Christian Boehning (University of Warwick)
2020-08-25 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/43/
Title: Rigid, not infinitesimally rigid surfaces of general type with ample canonical bundle (https://researchseminars.org/talk/ZAG/43/)
by Christian Boehning (University of Warwick) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In the talk I will report on work in progress, joint with Roberto Pignatelli and Hans-Christian von Bothmer, that concerns the construction of surfaces of general type with ample canonical bundle and Kuranishi space (and possibly also Gieseker moduli space) a non-reduced point. The main tools are configurations of lines and their incidence schemes as well as the theory of abelian covers due to Pardini and others.

Robert Berman (Chalmers University of Technology)
2020-08-27 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/44/
Title: Kahler-Einstein metrics, Archimedean Zeta functions and phase transitions (https://researchseminars.org/talk/ZAG/44/)
by Robert Berman (Chalmers University of Technology) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
While the existence of a unique Kahler-Einstein metrics on a canonically polarized manifold X was established already in the seventies there are very few explicit formulas available (even in the case of complex curves!). In this talk I will give a non-technical introduction to a probabilistic approach to Kahler-Einstein metrics, which, in particular, yields canonical approximations of the Kahler-Einstein metric on X. The approximating metrics in question are expressed as explicit period integrals and the conjectural extension to the case of a Fano variety leads to some intriguing connections with Zeta functions and the theory of phase transitions in statistical mechanics.

Jarod Alper (University of Washington)
2020-09-03 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/46/
Title: Advances in moduli theory (https://researchseminars.org/talk/ZAG/46/)
by Jarod Alper (University of Washington) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We will survey how recent advances in moduli theory allow for a new technique to construct projective moduli spaces of objects with potentially non-finite automorphism groups such as sheaves, complexes or Fano varieties. We will primarily explore this technique through the lens of the moduli space of vector bundles over a smooth curve where the definitions and concepts are most readily internalized. Time permitting, we will discuss applications to Bridgeland stability and perhaps how this construction technique, which works now only in characteristic zero, can be generalized to positive characteristic.

Andreas Höring (Université Côte d'Azur)
2020-09-08 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/47/
Title: Fano manifolds such that the tangent bundle is (not) big (https://researchseminars.org/talk/ZAG/47/)
by Andreas Höring (Université Côte d'Azur) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Let X be a Fano manifold. While the properties of the anticanonical divisor -KX and its multiples have been studied by many authors, the positivity of the tangent bundle TX is much more elusive. We give a complete characterisation of the pseudoeffectivity of TX for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds. This is joint work with Jie Liu and Feng Shao.

Joaquín Moraga (Princeton University)
2020-09-10 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/48/
Title: Topology and geometry of Kawamata log terminal singularities (https://researchseminars.org/talk/ZAG/48/)
by Joaquín Moraga (Princeton University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk, we will discuss the topology of Kawamata log terminal singularities. We show that from the perspective of the fundamental group klt singularities are close to quotient singularities. For instance, the regional fundamental group of a klt singularity of dimension n contains a normal abelian subgroup of rank at most n and index at most c(n). Then, we proceed to study geometric implications of the topology of klt singularities. We give a characterization theorem in the case that the abelian part of the fundamental group is large of full rank.

Julie Rana (Lawrence University)
2020-09-15 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/49/
Title: Singularities and divisors in the moduli space of surfaces (https://researchseminars.org/talk/ZAG/49/)
by Julie Rana (Lawrence University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The KSBA moduli space of stable surfaces (surfaces with slc singularities and ample canonical class) is a natural compactification of Gieseker's moduli space of surfaces of general type. In contrast with the moduli space of curves, very little is known about the birational geometry of KSBA moduli spaces; indeed, there are very few examples of divisors in KSBA moduli spaces. I will give an example of a divisor in the moduli space of quintic surfaces corresponding to surfaces with cyclic quotient singularities. I also discuss joint work with Giancarlo Urz\'ua where we give bounds that help to narrow the search.

Anthony Varilly-Alvarado (Rice University)
2020-05-27 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/50/
Title: Rational curves on K3 surfaces (https://researchseminars.org/talk/ZAG/50/)
by Anthony Varilly-Alvarado (Rice University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract: TBA

Jean-Louis Colliot-Thelene (Université Paris-Sud)
2020-09-22 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/51/
Title: Zero-cycles on del Pezzo surfaces (https://researchseminars.org/talk/ZAG/51/)
by Jean-Louis Colliot-Thelene (Université Paris-Sud) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Let k be an arbitary field of characteristic zero and X be a smooth, projective, geometrically rational surface. Birational classification of such surfaces (over k) is due to Enriques, Manin, Iskovskikh, Mori. We are interested in zero-cycles on such surfaces. In 1974, Daniel Coray showed that on a smooth cubic surface X with a closed point of degree prime to 3 there exists a closed point of degree 1, 4 or 10. Whether 4 and 10 may be omitted is still an open question. We first show how a combination of generisation, specialisation, Bertini theorems and "large" fields avoids considerations of special cases in Coray's argument. For smooth cubic surfaces X with a rational point, we show that any zero-cycle of degree at least 10 is rationally equivalent to an effective cycle. We establish analogues of these results for del Pezzo surfaces X of degree 2 and of degree 1. This completes the proof that for any geometrically rational surface X with a rational point, there exists an integer N which depends only on the geometry of the surface, such that any zero-cycle of degree at least N is rationally equivalent to an effective zero-cycle. For smooth cubic surfaces X without a rational point, we relate the question whether there exists a degree 3 point which is not on a line to the question whether rational points are dense on a del Pezzo surface of degree 1.

Alan Thompson (Loughborough University)
2020-09-24 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/52/
Title: Mirror symmetry for fibrations and degenerations (https://researchseminars.org/talk/ZAG/52/)
by Alan Thompson (Loughborough University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In a 2004 paper, Tyurin briefly hinted at a novel relationship between Calabi-Yau mirror symmetry and the Fano-LG correspondence. More specifically, if one can degenerate a Calabi-Yau manifold to a pair of (quasi-)Fanos, then one expects to be able to express the mirror Calabi-Yau in terms of the corresponding Landau-Ginzburg models. Some details of this correspondence were worked out by C. F. Doran, A. Harder, and I in a 2017 paper, but much remains mysterious. In this talk I will describe recent attempts to better understand this picture, and how it hints at a broader mirror symmetric correspondence between degeneration and fibration structures. As an example of this correspondence, I will discuss the question of finding mirrors to certain exact sequences which describe the Hodge theory of degenerations. The material in this talk is joint work in progress with C. F. Doran.

Yuri Zarhin (Penn State University)
2020-09-29 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/53/
Title: Jordan properties of automorphism groups of algebraic varieties and complex manifolds (https://researchseminars.org/talk/ZAG/53/)
by Yuri Zarhin (Penn State University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
A classical theorem of Jordan asserts that each finite subgroup of the complex general linear group GL(n) is "almost commutative": it contains a commutative normal subgroup with index bounded by an universal constant that depends only on n. We discuss an analogue of this property for the groups of birational (and biregular) automorphisms of complex algebraic varieties and the groups of bimeromorphic automorphisms of compact complex manifolds.

Burt Totaro (UCLA)
2020-10-01 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/54/
Title: The Hilbert scheme of points on affine space (https://researchseminars.org/talk/ZAG/54/)
by Burt Totaro (UCLA) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will discuss the Hilbert scheme of d points in affine n-space, with some examples. This space has many irreducible components for n at least 3 and has been poorly understood. For n greater than d, we determine the homotopy type of the Hilbert scheme in a range of dimensions. Many questions remain. (Joint with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Maria Yakerson.)

Junliang Shen (MIT)
2020-10-06 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/55/
Title: Cohomology of the moduli of Higgs bundles and the Hausel-Thaddeus conjecture (https://researchseminars.org/talk/ZAG/55/)
by Junliang Shen (MIT) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We describe the cohomological structure of the moduli space of stable SL_n Higgs bundles on a curve following the topological mirror symmetry conjecture of Hausel-Thaddeus. For the approach, we establish a connection between:
(a) the moduli space of twisted Higgs bundles by an effective divisor of degree greater than 2g-2, and
(b) the moduli space of K_C-Higgs bundles,
using vanishing cycle functors. This allows us to apply Ngo's support theorem, which has a simpler form in the case (a) (by Ngo, Chaudouard-Laumon, de Cataldo), to the case of (b) which concerns hyper-Kähler geometries. In particular, this gives a new proof of the Hausel-Thaddeus conjecture proven previously by Gröchenig-Wyss-Ziegler via p-adic integrations. Based on joint work with Davesh Maulik.

Timothy Logvinenko (Cardiff University)
2020-10-08 15:30 - 16:30 | https://researchseminars.org/talk/ZAG/56/
Title: Skein-triangulated representations of generalised braids (https://researchseminars.org/talk/ZAG/56/)
by Timothy Logvinenko (Cardiff University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Ordinary braid group Br_n is a well-known algebraic structure which encodes configurations of n non-touching strands (“braids”) up to continious transformations (“isotopies”). A classical result of Khovanov and Thomas states that there is a natural categorical action of Br_n on the derived category of the cotangent bundle of the variety of complete flags in C^n.
In this talk, I will introduce a new structure: the category GBr_n of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In the context of triangulated categories, it is natural to impose certain relations which result in the notion of a skein-triangulated representation of GBr_n.
A decade-old conjecture states that there a skein-triangulated action of GBr_n on the cotangent bundles of the varieties of full and partial flags in C^n. We prove this conjecture for n = 3. We also show that any categorical action of Br_n can be lifted to a skein-triangulated action of GBr_n, which behaves like a categorical nil Hecke algebra. This is a joint work with Rina Anno and Lorenzo De Biase.

Ana-Maria Castravet (Université de Versailles St-Quentin-en-Yvelines)
2020-10-13 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/57/
Title: Blown-up toric surfaces with non-polyhedral effective cone (https://researchseminars.org/talk/ZAG/57/)
by Ana-Maria Castravet (Université de Versailles St-Quentin-en-Yvelines) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will report on recent joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.
We construct examples of projective toric surfaces whose blow-up at a general point has a
non-polyhedral pseudoeffective cone, both in characteristic 0 and in prime characteristic.
As a consequence, we prove that the pseudo-effective cone of the Grothendieck-Knudsen moduli space of stable, n-pointed, rational stable curves, is not polyhedral if
n>=10 in characteristic 0 and in positive characteristic for an infinite set of primes of positive density.
In particular, these moduli spaces are not Mori dream spaces even in positive characteristic.

Sandra Di Rocco (KTH)
2020-10-15 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/58/
Title: Algebraic Geometry of Data (https://researchseminars.org/talk/ZAG/58/)
by Sandra Di Rocco (KTH) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
It is often convenient to visualise algebraic varieties (and hence systems of polynomial equations) by sampling. The key challenge is to have the right distribution and density in order to recover the shape, i.e the topology of the variety. Bottlenecks are pairs of points on the variety joined by a line which is normal to the variety at both points. These points play a special role in determining the appropriate density of a point-sample. Under suitable genericity assumptions the number of bottlenecks of an affine variety is finite and we call it the bottleneck degree. We show that it is determined by (classical) invariants of the variety, i.e. polar classes. The talk is based on joint work with D. Eklund and M. Weinstein.

Gergely Berczi (Aarhus University)
2020-10-20 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/59/
Title: Non-reductive group actions and hyperbolicity (https://researchseminars.org/talk/ZAG/59/)
by Gergely Berczi (Aarhus University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Non-reductive reparametrisation group actions play central role in hyperbolicity questions. Using recently developed intersection theory on non-reductive geometric invariant theory-type quotients and following the strategy of Demailly, Siu et al, last year we completed a proof of the Green-Griffiths-Lang and Kobayashi hyperbolicity conjectures for generic hypersurfaces of polynomial degree. We explain elements of the proof. Joint work with F. Kirwan.

David Mumford (Harvard University and Brown University)
2020-06-25 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/60/
Title: Q&A with legendary geometers: David Mumford (https://researchseminars.org/talk/ZAG/60/)
by David Mumford (Harvard University and Brown University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Q&A with David Mumford (please, no algebraic geometry questions). If you want to ask a question you should e-mail it in advance to i.cheltsov@ed.ac.uk

Christian Liedtke (Technical University of Munich)
2020-09-17 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/61/
Title: Rational curves on K3 surfaces (https://researchseminars.org/talk/ZAG/61/)
by Christian Liedtke (Technical University of Munich) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We prove that every complex projective K3 surface contains infinitely rational curves, which confirms a folklore conjecture on K3 surfaces. This was previously known for elliptic K3 surfaces (Bogomolov-Tschinkel), for very general K3 surfaces (Chen), as well as for K3 surfaces of odd Picard rank (Bogomolov-Hassett-Tschinkel, Li-Liedtke). We finish this conjecture by introducing two new techniques: “regeneration” (a sort of converse to degeneration) and the “marked point trick” (a technique for controlled degenerations), which allows to treat the missing cases. This is joint work with Xi Chen and Frank Gounelas.

Alexander Borisov (Binghamton University)
2020-10-22 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/62/
Title: Projective geometry approach to Jacobian Conjecture (https://researchseminars.org/talk/ZAG/62/)
by Alexander Borisov (Binghamton University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
acobian Conjecture is one of the oldest unsolved problems in Algebraic Geometry, going back to a 1939 paper by Keller. It is infamous for the large number of incorrect proofs that have been proposed over the years. In fact, it is quite possible that the conjecture is false, especially in higher dimensions. For the past 10-15 years I have been making slow but steady progress in understanding this enigma in dimension two, using classical methods of algebraic geometry of projective surfaces and some inspiration from the Minimal Model Program. I will explain my approach and where it has led me, and will also discuss some related conjectures.

Daniel Greb (University of Duisburg-Essen)
2020-10-27 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/63/
Title: Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor (https://researchseminars.org/talk/ZAG/63/)
by Daniel Greb (University of Duisburg-Essen) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will discuss a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties by Q-Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. I will further examine cases in which this stability condition is satisfied, comparing it to K-semistability and related notions. This is joint work with Stefan Kebekus and Thomas Peternell.

Junyan Cao (Université Côte d'Azur)
2020-10-29 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/64/
Title: On the Ohsawa-Takegoshi extension theorem (https://researchseminars.org/talk/ZAG/64/)
by Junyan Cao (Université Côte d'Azur) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Abstract: Since it was established, the Ohsawa-Takegoshi extension theorem turned out to be a fundamental tool in complex geometry. We establish a new extension result for twisted canonical forms defined on a hypersurface with simple normal crossings of a projective manifold with a control on its L^2 norme. It is a joint work with Mihai Paun.

Bianca Viray (University of Washington)
2020-11-03 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/65/
Title: On quadratic points on intersections of two quadrics (https://researchseminars.org/talk/ZAG/65/)
by Bianca Viray (University of Washington) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Springer's theorem and the Amer-Brumer theorem together imply that intersections of two quadrics have a rational point if and only if they have a 0-cycle of degree 1. In this talk, we consider whether this statement can be strengthened in the case when there is no rational point, namely when 1) the least degree of a 0-cycle can be 2, and 2) when this occurs, whether there is an effective 0-cycle of degree 2. We report on results in this direction, paying particular attention to the case of local and global fields. This is joint work with Brendan Creutz.

Cristiano Spotti (Aarhus University)
2020-11-05 13:00 - 14:00 | https://researchseminars.org/talk/ZAG/66/
Title: Geometric aspects of Kaehler-Einstein metrics on klt pairs (https://researchseminars.org/talk/ZAG/66/)
by Cristiano Spotti (Aarhus University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk I will discuss about the existence and geometric properties (e.g., tangent cones asymptotics, metric degenerations, etc...) of conical Kähler-Einstein metrics on klt pairs.

Aleksandr Pukhlikov (University of Liverpool)
2020-11-10 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/67/
Title: Rationally connected rational double covers of primitive Fano varieties (https://researchseminars.org/talk/ZAG/67/)
by Aleksandr Pukhlikov (University of Liverpool) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no Galois rational covers $X\dashrightarrow V$ with an abelian Galois group, where $X$ is a rationally connected variety. In particular, there are no rational maps $X\dashrightarrow V$ of degree 2 with $X$ rationally connected. This fact is true for many other families of primitive Fano varieties as well and motivates a conjecture on absolute rigidity of primitive Fano varieties.

‪Jürgen Hausen‬ (University of Tübingen)
2020-11-12 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/68/
Title: Automorphisms of k*-surfaces (https://researchseminars.org/talk/ZAG/68/)
by ‪Jürgen Hausen‬ (University of Tübingen) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
After recalling the necessary background on k*-surfaces, we give a complete description of the automorhpism group of a non-toric rational normal projective k*-surface in terms of isotropy group orders and self intersection numbers of suitable invariant curves. We also discuss the basic ingredients and ideas of the proof.

Michela Artebani (Universidad de Concepcion)
2020-11-17 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/69/
Title: Cox rings of K3 surfaces (https://researchseminars.org/talk/ZAG/69/)
by Michela Artebani (Universidad de Concepcion) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Given a normal complex projective variety X with finitely generated divisor class group, its Cox ring R(X) is the Cl(X)-graded algebra whose homogeneous pieces are Riemann-Roch spaces of divisors of X. This object is particularly interesting when it is finitely generated, since in such case X can be obtained as a GIT quotient of an open subset of Spec R(X) by the action of a quasi-torus [1]. Finding a presentation or even a minimal generating set for R(X) is in general a difficult problem, already in the case of surfaces. In this talk, after an introduction to the subject, we will concentrate on complex projective K3 surfaces, which are known to have finitely generated Cox ring exactly when their automorphism group is finite [2]. We show that the Cox ring can be generated by homogeneous elements whose degrees are either classes of (-2)-curves, sums of at most three elements in the Hilbert basis of the nef cone, or classes of divisors of the form 2(E+E'), where E,E' are elliptic curves with E.E'=2. As an application, we compute Cox rings of Mori dream K3 surfaces of Picard number 3 and 4. This is joint work with C. Correa Deisler, A. Laface and X. Roulleau [3,4].

References.
[1] I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015.
[2] M. Artebani, J. Hausen, and A. Laface, On Cox rings of K3 surfaces, Compos. Math. 146 (2010), no. 4, 964–998. arXiv:0901.0369
[3] M. Artebani, C. Correa Deisler, and A. Laface, Cox rings of K3 surfaces of Picard number three, J. Algebra 565C (2021), 598–626. arXiv:1909.01267
[4] M. Artebani, C. Correa Deisler, and X. Roulleau, Mori dream K3 surfaces of Picard number four: projective models and Cox rings. arXiv:2011.00475.

Yukari Ito (Kavli IPMU)
2020-11-19 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/70/
Title: The McKay correspondence (https://researchseminars.org/talk/ZAG/70/)
by Yukari Ito (Kavli IPMU) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The original McKay correspondence was observed by John McKay as a correspondence between a finite subgroup G of SL(2,C) and simple Lie algebra in representation theory and developed as a correspondence between the group G and the minimal resolution of the quotient singularity C^2/G in algebraic geometry. In this talk, I will introduce the McKay correspondence in dimension three and show recent progress and open problems.

Kazushi Ueda (University of Tokyo)
2020-11-24 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/71/
Title: Noncommutative del Pezzo surfaces (https://researchseminars.org/talk/ZAG/71/)
by Kazushi Ueda (University of Tokyo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Abstract: We introduce the notion of noncommutative del Pezzo surfaces, and show that a collection of 12-d general vector bundles of certain ranks and degrees on an elliptic curve produces a noncommutative del Pezzo surface of degree d. We also define the moduli stack of marked noncommutative del Pezzo surfaces, and show that it contains the configuration space of 9-d points in general position on the projective plane as a locally closed substack. This is a joint work in progress with Tarig Abdelgadir and Shinnosuke Okawa.

Yuji Odaka (Kyoto University)
2020-11-26 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/72/
Title: On compactifying moduli and degenerations of K-trivial varieties (https://researchseminars.org/talk/ZAG/72/)
by Yuji Odaka (Kyoto University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Some background review: the KSBA moduli of varieties of ample canonical classes is interpreted via K-stability resp., KE metrics (O’10, resp., Berman-Guenancia’13). A recent trend since 2012 is to establish its Fano analogue, and study their K-stability itself, which still continues to be developed by more and more contributors wonderfully. Luckily, in both cases, K-polystable / KE varieties (should) form projective (compact) moduli schemes.
However, nevertheless of general K-moduli expectation, such existence of projective moduli of K-polystable/cscK (polarized) varieties is NOT true “at the boundary”, even for classical K-trivial / Calabi-Yau cases. Indeed, as a general theory, no “canonical” algebro-geometric compactification theory of moduli of polarized CY vars seems established. E.g. An idea pursued and fairly developed is to attach ample extra divisors on the CY vars (to pass to “K:ample”-like situations) and take their “log KSBA” compactifications, but different choice of the extra divisors can lead to different log KSBA compactifications.
In our talk, based on our several recent papers (partially j.w.w. Yoshiki Oshima), we discuss the possibilities of still getting “canonical (geometric) compactifications” of the moduli of polarized K-trivial / CY varieties and corresponding "canonical limits", especially giving more explicit conjectures in hyperKahler / K3 case, with certain confirmations. This involves not only classical AG but also DG of collapsing CY metrics, symmetric space theory (Lie, Cartan, .. Satake..), non-archimedean/tropical geometry, and some mirror symmetric phenomena. Examples and pictures will be used for the illustration.

Takuzo Okada (Saga University)
2020-12-01 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/73/
Title: K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces (https://researchseminars.org/talk/ZAG/73/)
by Takuzo Okada (Saga University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
For Fano varieties, birational superrigidity and K-stability are completely different notions. On the other hand, they are both related to mildness of singularities of (pluri-)anticanonical divisors/systems. In fact, it is conjectured that a birational superrigid Fano variety is K-stable. In my talk I will explain recent results that give a positive answer to the conjecture for Fano 3-fold weighted hypersurfaces. This is a joint work with In-Kyun Kim and Joonyeong Won.

Hulya Arguz (Versailles Saint-Quentin-en-Yvelines University)
2020-12-03 13:00 - 14:00 | https://researchseminars.org/talk/ZAG/74/
Title: Enumerating punctured log Gromov-Witten invariants from wall-crossing (https://researchseminars.org/talk/ZAG/74/)
by Hulya Arguz (Versailles Saint-Quentin-en-Yvelines University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Punctured log Gromov—Witten invariants of Abramovich—Chen--Gross—Siebert are obtained by counting of stable maps with prescribed tangency conditions (which are allowed to be negative) relative to a not necessarily smooth divisor. In this talk we describe an algorithmic method to compute punctured log Gromov-Witten invariants of log Calabi-Yau varieties, which are obtained by blowing-up of toric varieties along hypersurfaces on the toric boundary. For this we use tropical geometry and wall-crossing computations. This is joint work with Mark Gross (arxiv:2007.08347).

Keiji Oguiso (University of Tokyo)
2020-12-08 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/75/
Title: Smooth projective rational varieties with non-finitely generated discrete automorphism group (https://researchseminars.org/talk/ZAG/75/)
by Keiji Oguiso (University of Tokyo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract: TBA

Artan Sheshmani (Harvard University Center for Mathematical sciences and Applications)
2020-12-10 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/76/
Title: DT invariants from Gerstenhaber-BV structures, and degeneration technique (https://researchseminars.org/talk/ZAG/76/)
by Artan Sheshmani (Harvard University Center for Mathematical sciences and Applications) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface. We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3. There are connections between these invariants and birational properties of the base surface which we will discuss if time permits.

Sho Tanimoto (Kumamoto University)
2020-12-15 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/77/
Title: Rational curves on Fano threefolds (https://researchseminars.org/talk/ZAG/77/)
by Sho Tanimoto (Kumamoto University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Mori proved that a smooth Fano variety contains a lots of rational curves using famous Bend and Break technique. Thus it is natural to study the space of rational curves on a smooth Fano variety. Lines and conics on Fano threefolds are well studied, and one may ask what one can say about higher degree rational curves. Recently we established Movable Bend and Break for Fano threefolds claiming that any free curve of high degree breaks into the union of two free curves. A proof is intricate, and it relies on many properties of three dimensional MMP such as Mori’s classification of divisorial contractions on smooth projective threefolds. In this talk I would like to explain some aspects of our proof of Movable Bend and Break as well as an application to Batyrev’s conjecture predicting a polynomial growth of the number of components of bounded degree for the moduli space of rational curves. If time permits, then I also explain a relation of our study to Geometric Manin’s conjecture which is an inspiration of our study. This is joint work with Roya Beheshti, Brian Lehmann, and Eric Riedl.

Andrew Harder (Lehigh University)
2020-12-17 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/78/
Title: Log symplectic pairs and mixed Hodge structures (https://researchseminars.org/talk/ZAG/78/)
by Andrew Harder (Lehigh University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
A log symplectic pair is a pair (X,Y) consisting of a smooth projective variety X and a divisor Y in X so that there is a non-degenerate log 2-form on X with poles along Y. I will discuss the relationship between log symplectic pairs and degenerations of hyperkaehler varieties, and how this naturally leads to a class of log symplectic pairs called log symplectic pairs of "pure weight". I will talk about common properties of cohomology rings of log symplectic pairs of pure weight and type III degenerations of hyperkaehler varieties, in particular, the fact that both have the curious hard Lefschetz' (CHL) property discovered by Hausel and Rodriguez-Villegas. Finally I will discuss partial results towards proving that in both of these cases, the CHL property is a consequence of P=W type results. Part of this is based on work with Li, Shen, and Yin.

Han-Bom Moon (Fordham University)
2020-12-22 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/79/
Title: Point configurations, phylogenetic trees, and dissimilarity vectors (https://researchseminars.org/talk/ZAG/79/)
by Han-Bom Moon (Fordham University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In 2004 Pachter and Speyer introduced the dissimilarity maps for phylogenetic trees and asked two important questions about their relationship with tropical Grassmannian. Multiple authors answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. In this talk, we present a weighted variant of the dissimilarity map and show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter-Speyer envisioned. This tropical variety has a geometric interpretation in terms of point configurations on rational normal curves. This is joint work with Alessio Caminata, Noah Giansiracusa, and Luca Schaffler.

Hokuto Uehara (Tokyo Metropolitan University)
2020-12-24 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/80/
Title: Exceptional collections on the Hilzebruch surface of degree 2 (https://researchseminars.org/talk/ZAG/80/)
by Hokuto Uehara (Tokyo Metropolitan University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The purpose of my talk is to clarify the structure of exceptional collections of the derived category of coherent sheaves on the Hirzebruch surface of degree 2 (a special weak del Pezzo surface), to the extent it is understood by Orlov and Kuleshov for del Pezzo surfaces.
(1) First, we prove that for any exceptional object in it, one can find an autoequivalence which sends it to an exceptional vector bundle.
This result was conjectured by Shinnosuke Okawa and the speaker in 2015.
(2) Refining the method of the proof of the above result, and based on a deformation argument, we prove that the braid group on 4 strands acts transitively (up to shifts) on the set of exceptional collections of length 4. This is a special case of an old conjecture by Bondal and Polishchuk.
(3) We also prove that any exceptional collection can be extended to a full exceptional collection.
My talk is based on a joint work with Shinnosuke Okawa (Osaka) and Akira Ishii (Nagoya).

Jingjun Han (Johns Hopkins University)
2020-12-29 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/81/
Title: The ACC for local volumes and boundedness of singularities (https://researchseminars.org/talk/ZAG/81/)
by Jingjun Han (Johns Hopkins University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Kawamata log terminal (klt) singularities form an important class of singularities due to its fundamental roles in MMP, K\”ahler-Einstein geometry, and K-stability. Recently, Chi Li introduced a new invariant called the local volume of a klt singularity which encodes lots of interesting geometric and topological information. A folklore conjecture predicts that local volumes satisfy the ascending chain condition (ACC) when the coefficients of the boundary divisors belong to a DCC set. In this talk, we will show the ACC conjecture for local volumes holds when the ambient variety is fixed. We will also explore the relation between log canonical thresholds, local volumes, and delta-plt blow-ups. This is based on ongoing joint work with Yuchen Liu and Lu Qi.

Hiromichi Takagi (Gakushuin University)
2021-01-07 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/84/
Title: Key varieties for prime Q-Fano threefolds of codimension 4 (https://researchseminars.org/talk/ZAG/84/)
by Hiromichi Takagi (Gakushuin University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I talk about my construction of key varieties for prime Q-Fano threefolds of codimension 4 related with (P^2)^2-fibrations.I discuss their relations with the cluster variety constructed by Coughlan and Ducat.

Patricio Gallardo (UC Riverside)
2021-01-12 18:00 - 19:00 | https://researchseminars.org/talk/ZAG/85/
Title: Compactifications of the moduli space of cubic surfaces (https://researchseminars.org/talk/ZAG/85/)
by Patricio Gallardo (UC Riverside) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We discuss the interplay between geometric and Hodge theoretical compactifications for the moduli space of cubic surfaces. In particular, we prove that Naruki's compactification is toroidal and has a modular interpretation in terms of stable pairs. This last is joint work with Matt Kerr and Luca Schaffler. If time allows, we will describe open questions and ongoing generalizations of such a relationship to the case of pairs involving cubic surfaces and their anticanonical divisors.

Hiromu Tanaka (University of Tokyo)
2021-01-14 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/86/
Title: On Mori fibre spaces in positive characteristic (https://researchseminars.org/talk/ZAG/86/)
by Hiromu Tanaka (University of Tokyo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The minimal model program conjecture predicts that any algebraic variety is birational to either a minimal model or a Mori fibre space.
In this talk, we first summarise some results on Mori fibre spaces in positive characteristic.
We also discuss some open problems related to this topic.

Shihoko Ishii (University of Tokyo)
2021-01-19 09:00 - 10:00 | https://researchseminars.org/talk/ZAG/87/
Title: Uniform bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds (https://researchseminars.org/talk/ZAG/87/)
by Shihoko Ishii (University of Tokyo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In the talk I will show that the minimal log discrepancy of every pair consisting of a smooth 3-fold and a "general" real ideal is computed by the divisor obtained by at most two weighted blow ups.

Our proof suggests the following conjecture:

Every pair consisting of a smooth N-fold and a ``general” real ideal is computed by a divisor obtained by at most N-1 weighted blow ups.

This is regarded as a weighted blow up version of Mustata-Nakamura’s conjecture.

Osamu Fujino (Osaka University)
2021-01-21 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/88/
Title: On extremal contractions of log canonical pairs (https://researchseminars.org/talk/ZAG/88/)
by Osamu Fujino (Osaka University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The cone and contraction theorem holds for projective log canonical pairs. Let us consider an extremal contraction morphism of a log canonical pair. We prove that every irreducible component of the exceptional locus is uniruled. This result was first proved by Yujiro Kawamata for kawamata log terminal pairs. His proof uses a relative Kawamata--Viehweg vanishing theorem for projective bimeromorphic morphisms of complex analytic spaces and does not work for log canonical pairs. Our approach is based on the theory of quasi-log schemes and can be applied to more general settings. We need a semipositivity theorem coming from the theory of variations of mixed Hodge structure. We note that we do not use the minimal model program.

Frances Kirwan (University of Oxford)
2021-01-26 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/89/
Title: Moduli of unstable objects in algebraic geometry (https://researchseminars.org/talk/ZAG/89/)
by Frances Kirwan (University of Oxford) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT), developed in the 1960s; many other moduli spaces of 'stable' objects can be constructed using GIT, as well as in other ways. The aim of this talk is to explain how recent methods from a version of GIT for non-reductive group actions can help us to use suitable 'stability conditions' to stratify moduli stacks into locally closed strata such that not only the open 'stable' strata but also the 'unstable' strata have coarse moduli spaces. In the case of moduli stacks of bundles over a nonsingular projective curve, these stratifications refine the stratification by Harder-Narasimhan type. The talk is based on joint work with Gergely Berczi, Vicky Hoskins and Joshua Jackson.

Fabrizio Catanese (University of Bayreuth)
2021-01-28 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/90/
Title: Varieties of nodal surfaces, Coding theory, and cubic discriminants (https://researchseminars.org/talk/ZAG/90/)
by Fabrizio Catanese (University of Bayreuth) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Nodal Hypersurfaces Y in projective space are those whose singularities have nondegenerate Hessian.
Basic numerical invariants are the dimension n and the degree d of the hypersurface Y, and the number \nu of singular points.
If you fix those integers (n, d,\nu) these hypersurfaces are parametrized by the so-called Nodal Severi varieties F(n, d, \nu).
The first basic questions concerning them are:
1) for which triples is F(n, d, \nu) nonempty ?
2) When is it irreducible ?

Already intriguing is the situation for surfaces, indeed for n=2 the answer to 1) is known for d <= 6, also the maximal number of nodes \mu (d) that a nodal surface in 3-space of degree d can have is known only for d <= 6.

The known maximizing nodal surfaces (those with \mu(d) nodes) are: the Cayley cubic, the Kummer quartic surfaces, the Togliatti quintics, the Barth sextic.

An important chapter in Coding theory is the theory of binary linear codes, vector subspaces of a vector space (Z/2)^n.

I will recall basic notions and methods of coding theory (e.g. the McWilliams identities) and describe some codes related to quadratic forms.

Nodal surfaces are related to coding theory via the first homology of their smooth part: it is a binary code K, which was used by Beauville to show that, for d=5 , \mu(d) = 31. Coding theory was also crucial in order to prove that \mu(6) < = 65.

Our main results concern the cases d = 4,5,6 (d=2,3 being elementary).

THM 1. For d=4 the components of F(4, \nu) and their incidence correspondence are determined by their extended codes K’, which are all the shortenings of the first Reed Muller code.

We extend this result to nodal K3 surfaces of all degrees, this sheds light on the case d=5.

THM 2. For d=5 the codes K occurring are classified, up to a possible exception. F(5, \nu) is irreducible for \nu = 31, and for \nu = 29,30,31 these surfaces are discriminants of the projection of a cubic hypersurface in 5-space.

THM 3. For d=6 and \nu = 65 the codes K, K’ are uniquely determined, and can be described explicitly via the Doro-Hall graph, attached to the group \SigmaL(2, 25), and the geometry of the Barth sextic. Every 65 nodal sextic occurs as discriminant of the projection of a cubic hypersurface in 6-space with < = 33 nodes.

Irreducibility for d=6, and 65 nodes, is related to the geometry of nodal cubic hypersurfaces in n-space, and of the linear subspaces contained in them.

We pose the question whether, in the case of even dimension n, the cubic hypersurface with maximal number of singularities is
projectively equivalent to the Segre cubic s_1=s_3=0 (which is locally rigid).

For theorem 2 I benefited of the cooperation of Sandro Verra, for theorem 3 of Yonghwa Cho, Michael Kiermaier, Sascha Kurz and the Linux Cluster of the Universitaet Bayreuth, while Davide Frapporti and Stephen Coughlan cooperated for the geometry of nodal cubic hypersurfaces.

Simon Donaldson (Imperial College London and Simons Center for Geometry and Physics)
2021-02-02 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/91/
Title: Enumerative geometry, Fredholm analysis and moduli spaces of surfaces of general type (https://researchseminars.org/talk/ZAG/91/)
by Simon Donaldson (Imperial College London and Simons Center for Geometry and Physics) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In the first part of the talk we will review some background in deformation theory, comparing the points of view from algebraic geometric and differential geometry/global analysis. We will review in outline some known established examples in which a "virtual fundamental class" of a moduli space can be defined. In the second part of the talk we will explore the possibility that these ideas can be applied to moduli spaces of surfaces of general type using the KSBA compactification. We will make some standard observations about these moduli spaces, whose dimension often differs from the virtual dimension. We will illustrate the discussion by a calculation in the case of sextic surfaces with a particular finite symmetry group.

Shinnosuke Okawa (Osaka University)
2021-02-04 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/92/
Title: Moduli space of semiorthogonal decompositions (https://researchseminars.org/talk/ZAG/92/)
by Shinnosuke Okawa (Osaka University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Semiorthogonal decomposition (SOD) of triangulated categories is quite interesting and of fundamental importance for various reasons. For example, SOD of the derived category of coherent sheaves is closely related to the geometry of varieties, such as the minimal model program (MMP) among others. It is therefore desirable to understand the general properties of SODs, partly so as to classify SODs of as many triangulated categories as possible. The purpose of this talk is to explain certain moduli spaces of SODs which we introduced. To a smooth projective morphism of excellent schemes f: X \to B, we associate an algebraic space over B which classifies the SODs of the derived categories of the fibers of f. We will discuss properties and various aspects of this moduli space including applications, comparison to MMP, and open problems.

Enrica Floris (Université de Poitiers)
2021-02-09 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/93/
Title: Connected algebraic groups acting on Fano fibrations over $\mathbb P^1$ (https://researchseminars.org/talk/ZAG/93/)
by Enrica Floris (Université de Poitiers) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Let G be a connected algebraic group and X a variety endowed with a regular action of G and a Mori fibre space X/P1 whose fibre is a Fano variety of Picard rank at least 2. In this talk I will explain why there is a proper horizontal subvariety of X which is invariant under the action of G, alongside with some applications of this result to the classification of connected algebraic subgroups of the Cremona group in dimension 4. This is a joint work with Jeremy Blanc.

Tomasso de Fernex (University of Utah)
2021-02-11 18:00 - 19:00 | https://researchseminars.org/talk/ZAG/94/
Title: Motivic integration on Berkovic spaces (https://researchseminars.org/talk/ZAG/94/)
by Tomasso de Fernex (University of Utah) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The purpose of the talk is to offer a new perspective on motivic integration. Working over a field with trivial norm, I will introduce a motivic measure on the Berkovich analytification of an algebraic variety and define integration in this setting. The construction is geometric with a similar spirit as Kontsevich’s original definition, and leads to the formulation of a functorial theory which mirrors, in this aspect, the functoriality of Cluckers and Loeser's approach via constructible motivic functions. The approach does not rely on model theory but rather on geometric properties such as resolution of singularities and the weak factorization theorem. The talk is based on joint work with Chung Ching Lau.

Yusuke Nakamura (University of Tokyo)
2021-02-16 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/95/
Title: Inversion of adjunction for quotient singularities (https://researchseminars.org/talk/ZAG/95/)
by Yusuke Nakamura (University of Tokyo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The minimal log discrepancy is an invariant of singularities defined in birational geometry, and it is related to the conjecture of termination of flips. In this talk, we will discuss the minimal log discrepancies of quotient singularities. I will show that the PIA (precise inversion of adjunction) conjecture holds for quotient singularities. The main tool of this talk involves the theory of the arc space of a quotient singularity established by Denef and Loeser. This is joint work with Kohsuke Shibata.

Frediani Paola (Università di Pavia)
2021-02-18 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/96/
Title: A canonical Hodge theoretic projective structure on compact Riemann surfaces (https://researchseminars.org/talk/ZAG/96/)
by Frediani Paola (Università di Pavia) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk we will show the existence of a canonical projective structure on every compact Riemann surface, coming from Hodge theory. We will show that it differs from the canonical projective structure produced by the uniformisation theorem. In fact the (0,1)- component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves are different. The one corresponding to the projective structure coming from uniformisation was computed by Zograf and Takhtadzhyan as the Weil-Petersson Kaehler form on the moduli space of curves. Ours is the pullback via the Torelli map of a nonzero constant scalar multiple of the Siegel form on the moduli space of principally polarised abelian varieties. These are results obtained in collaboration with I. Biswas, E. Colombo and G.P. Pirola.

Shing-Tung Yau (Harvard University)
2021-02-23 15:30 - 16:30 | https://researchseminars.org/talk/ZAG/97/
Title: Geometry of Conifold Transitions (https://researchseminars.org/talk/ZAG/97/)
by Shing-Tung Yau (Harvard University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Conifold transitions were introduced by Clemens, Reid and Friedman to connect Calabi-Yau threefolds with different topologies. However, this operation may produce a complex manifold with trivial canonical bundle which is non-Kahler. I will discuss this transition from the point of view of metrics and differential geometry, and propose a non-Kahler analog of Calabi-Yau metrics which originates in heterotic string theory. This talk will contain joint works with T.C. Collins, J.-X. Fu, J. Li, and S. Picard.

Atsushi Ito (Nagoya University)
2021-02-25 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/98/
Title: Linear systems on abelian varieties via M-regularity of Q-twisted sheaves (https://researchseminars.org/talk/ZAG/98/)
by Atsushi Ito (Nagoya University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
For an ample line bundle $L$ on an abelian variety $X$, it is known that $L^n$ is basepoint free if $n \geq 2$, projectively normal if $n \geq 3$, and the ideal of $X$ embedded by $|L^n|$ is generated by quadrics if $n \geq 4$. As a generalization of these results, Lazarsfeld conjectures that $L^n$ satisfies property $(N_p)$ if $n \geq p+3$.
This conjecture is affirmatively proved by Pareschi and strengthen by Pareschi-Popa by the theory of M-regularity. Recently, Jiang and Pareschi consider (variants of) M-regularity of $\mathbb{Q}$-twisted sheaves and it turns out that this is very useful when we study the linear system $|L|$ of $L$ itself, not only $L^n$ for $n \geq 2$. In this talk, I will explain this topic and some recent results.

Fernando Cukierman (University of Buenos Aires)
2021-03-02 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/99/
Title: Deformations of exterior differential ideals and applications (https://researchseminars.org/talk/ZAG/99/)
by Fernando Cukierman (University of Buenos Aires) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The main theme of this talk is the geometry of moduli spaces of algebraic singular foliations on smooth projective algebraic varieties over the complex numbers. A motivating problem is the determination of the irreducible components of such moduli spaces. We plan to discuss some basic facts on deformations of exterior differential ideals. With these tools we study deformations of several types of Pfaff ideals, obtaining some new irreducible components of spaces of singular foliations of any codimension. This is based on joint work with Cesar Massri.

Kiwamu Watanabe (Chuo University)
2021-03-04 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/100/
Title: Positivity of the exterior power of the tangent bundles (https://researchseminars.org/talk/ZAG/100/)
by Kiwamu Watanabe (Chuo University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
By Mori's solution of the Hartshorne conjecture, the only smooth projective variety with ample tangent bundle is the projective space. As a generalization of the Hartshorne conjecture, Demially, Peternell and Schneider studied smooth projective varieties with nef tangent bundle. They proved that such variety X admits an étale cover Y such that the Albanese map Y \to Alb(Y) is a smooth morphism whose fibers are smooth Fano varieties with nef tangent bundle. In this talk, we will study smooth projective varieties such that the r-th exterior power of the tangent bundle is nef, paying special attention to the case r=2. This talk is based on the paper arXiv:2011.01427.

Shunsuke Takagi (University of Tokyo)
2021-03-09 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/101/
Title: Deformations of F-pure and F-regular singularities (https://researchseminars.org/talk/ZAG/101/)
by Shunsuke Takagi (University of Tokyo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
F-regular and F-pure singularities are singularities in positive characteristic defined in terms of Frobenius splitting properties. Ma and Schwede recently proved that a normal Q-Gorenstein complex variety has log terminal singularities if its reduction modulo p has F-regular singularities for a single prime p. I will discuss the analog of their result for log canonical singularities. I will also explain how F-regular singularities behave under equal and mixed characteristic deformations. This talk is based on joint work with Kenta Sato.

Carolina Araujo (IMPA)
2021-03-11 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/102/
Title: Birational geometry of Calabi-Yau pairs and 3-dimensional Cremona transformations (https://researchseminars.org/talk/ZAG/102/)
by Carolina Araujo (IMPA) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Abstract: Recently, Oguiso addressed the following question, attributed to Gizatullin: ``Which automorphisms of a smooth quartic K3 surface $D\subset \mathbb{P}^3$ are induced by Cremona transformations of the ambient space $\mathbb{P}^3$?'' When $D\subset \mathbb{P}^3$ is a quartic surface, $(\mathbb{P}^3,D)$ is an example of a \emph{Calabi-Yau pair}, that is, a pair $(X,D)$ consisting of a normal projective variety $X$ and an effective Weil divisor $D$ on $X$ such that $K_X+D\sim 0$. In this talk, I will explain a general framework to study the birational geometry of mildly singular Calabi-Yau pairs. Then I will focus on the case of singular quartic surfaces $D\subset \mathbb{P}^3$. Our results illustrate how the appearance of increasingly worse singularities in $D$ enriches the birational geometry of the pair $(\mathbb{P}^3, D)$, and lead to interesting subgroups of the Cremona group of $\mathbb{P}^3$. This is a joint work with Alessio Corti and Alex Massarenti.

Taro Sano (Kobe University)
2021-03-16 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/103/
Title: Birational boundedness of some Calabi-Yau hypersurfaces (https://researchseminars.org/talk/ZAG/103/)
by Taro Sano (Kobe University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
It is well-known that complex projective K3 surfaces are connected by analytic deformations, but they are algebraically unbounded. Nevertheless, Reid, Iano-Fletcher and Kollar-Johnson showed the finiteness of weighted Calabi-Yau hypersurfaces. Motivated by this, we study plt Calabi-Yau pairs (X,D) and show finiteness of D in some cases. In particular, we show that anticanonical K3 surfaces form a birationally bounded family. We also exhibit examples of K3 surfaces of a fixed degree whose birational contractions form an unbounded family, thus the birational boundedness is optimal in a sense.

Chris Lazda (University of Warwick)
2021-03-18 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/104/
Title: A Neron-Ogg-Shafarevich criterion for K3 surfaces (https://researchseminars.org/talk/ZAG/104/)
by Chris Lazda (University of Warwick) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The naive analogue of the Néron-Ogg-Shafarevich criterion fails for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified etale cohomology groups, but which do not admit good reduction over K. Assuming potential semi-stable reduction, I will show how to correct this by proving that a K3 surface has good reduction if and only if is second cohomology is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain “canonical reduction” of X. This is joint work with B. Chiarellotto and C. Liedtke.

John Lesieutre (Pennsylvania State University)
2021-03-23 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/105/
Title: Rational surface automorphisms without periodic curves (https://researchseminars.org/talk/ZAG/105/)
by John Lesieutre (Pennsylvania State University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Although there are many examples known of infinite-order automorphisms of rational surfaces, in most cases these automorphisms have at least one periodic curve C (a curve for which f^n(C) = C for some n). I will explain the construction of a class of rational surfaces which admit a large group of automorphisms with no invariant curves. The example makes use of several classical constructions, and the surfaces admit contructions down to three different Coble surfaces.

Giulia Sacca (Columbia University)
2021-03-25 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/106/
Title: Fano manifolds associated to hyperkahler manifolds (https://researchseminars.org/talk/ZAG/106/)
by Giulia Sacca (Columbia University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
It is known that to some Fano manifolds whose cohomology looks like that of a K3 surface, one can associate, via geometric constructions, examples of hyperkahler manifolds. In this talk I will report on the first steps of a program whose aim is to reverse this construction: starting from a hyperkahler manifold how to recover geometrically a Fano manifold? This is joint work with L. Flapan, E. Macri, and K. O'Grady.

Katsuhisa Furukawa (Josai University)
2021-03-30 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/107/
Title: On the singular loci of higher secant varieties of Veronese embeddings (https://researchseminars.org/talk/ZAG/107/)
by Katsuhisa Furukawa (Josai University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
For a projective variety X in P^N, the k-secant variety \sigma_k(X) is defined to be the closure of the union of k-planes in P^N spanned by k-points of X. It is well known that \sigma_{k-1}(X) is contained in the singular locus of \sigma_k(X). Let us consider the case when X is the image of the Veronese embedding P^n to P^N of degree d, where N = \binom{d+N}{d}-1. In the case of k=3, K. Han showed that \Sing(\sigma_3(X)) = \sigma_2(X), except when d=4 and n > 2. In the exceptional case, \Sing(\sigma_3(X)) is the union of \sigma_2(X) and D, where D is an irreducible subset. In this talk, we first give a geometric description of this D for k = 3, and next study the case of k > 3. In particular, I will explain projective techniques with respect to an explicit calculation of the Gauss map of X and the projection from the incidence correspondence of \sigma_k(X). This is a joint work with Kangjin Han.

Vyacheslav Shokurov (Johns Hopkins University)
2021-04-01 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/108/
Title: Around a-n-complements (https://researchseminars.org/talk/ZAG/108/)
by Vyacheslav Shokurov (Johns Hopkins University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
A general conjecture about a-n-complements will be discussed in connection with strict $\delta$-lc singularities, generalized pairs and McKernan conjecture.

Takehiko Yasuda (Osaka University)
2021-04-06 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/109/
Title: Stringy motives and local fundamental groups of klt surface singularities in arbitrary characteristic (https://researchseminars.org/talk/ZAG/109/)
by Takehiko Yasuda (Osaka University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk, I will speak about an application of stringy motives to local fundamental groups of klt surface singularities. Xu proved that klt singularities in characteristic zero have finite local fundametal groups. I will explain how to prove that the same is true in arbitrary characteristic and in dimension two, which had been unknown in characteristics two and three. The key of the proof is to study the behavior of stringy motives under quasi-etale Galois covers of klt singularities. This is a joint work with Javier Carvajal-Rojas.

Adrian Langer (Institute of Mathematics, University of Warsaw, Poland)
2021-04-08 13:00 - 14:00 | https://researchseminars.org/talk/ZAG/110/
Title: Chern classes of vector bundles (https://researchseminars.org/talk/ZAG/110/)
by Adrian Langer (Institute of Mathematics, University of Warsaw, Poland) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will talk about various restrictions on Chern classes of vector bundles on algebraic varieties. One of the most important is Bogomolov's inequality saying that the degree of the discriminant of a semistable vector bundle on a smooth complex algebraic variety is non-negative. The degree zero case essentially corresponds to flat vector bundles. There are also versions of this inequality for Higgs bundles and in positive characteristic. I will talk about some applications of Bogomolov's inequality and its possible variants in the Chow ring of the variety.

Karl Schwede (University of Utah)
2021-04-13 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/111/
Title: Recent progress in mixed characteristic higher dimensional algebraic geometry (https://researchseminars.org/talk/ZAG/111/)
by Karl Schwede (University of Utah) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In characteristic zero birational algebraic geometry, Kawamata-Viehweg vanishing is a centrally important tool. For some applications in characteristic p > 0, one may use Frobenius and perturbations as a replacement for resolution of singularities and Kawamata-Viehweg vanishing. This talk will show how to use Bhatt's vanishing theorem for absolute integral closures mixed characteristic as a replacement for resolutions and Kawamata-Viehweg vanishing theorems in a number of applications. This is joint work with B. Bhatt, L. Ma, Z. Patakfalvi, K. Tucker, J. Waldron and J. Witaszek.

Yasunari Nagai (Waseda University)
2021-04-15 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/112/
Title: Rational normal quintic curves on a cubic fourfold (https://researchseminars.org/talk/ZAG/112/)
by Yasunari Nagai (Waseda University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We study the moduli of rational normal quintic curves on a cubic fourfold and its compactifications with a view toward projective symplectic geometry. This is a work in progress with Manfred Lehn and Duco van Straten.

Jakub Witaszek (University of Michigan)
2021-04-20 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/113/
Title: Relative semiampleness in mixed characteristic. (https://researchseminars.org/talk/ZAG/113/)
by Jakub Witaszek (University of Michigan) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk we will discuss the existence of contractions and base point freeness of line bundles in mixed characteristic in the context of the arithmetic Minimal Model Program.

Sandor Kovacs (University of Washington)
2021-04-22 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/114/
Title: Hodge sheaves for singular families (https://researchseminars.org/talk/ZAG/114/)
by Sandor Kovacs (University of Washington) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
This is a report on joint work with Behrouz Taji. Given a flat projective morphism [f:X\to B] of complex varieties, assuming that [B] is smooth, we construct a system of reflexive Hodge sheaves on [B] . If in addition [X] is also smooth then this system gives an extension of the Hodge bundle underlying the VHS of the smooth locus of [f] . This in turn provides a criterion that all VHSs of geometric origin must satisfy. As an independent application we prove a singular version of Viehweg's conjecture about base spaces of families of maximal variation.

Yuri Tschinkel (Courant Institute of Mathematical Sciences, New York University)
2021-04-27 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/115/
Title: Equivariant birational types (https://researchseminars.org/talk/ZAG/115/)
by Yuri Tschinkel (Courant Institute of Mathematical Sciences, New York University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will discuss joint work with A. Kresch and B. Hassett concerning new invariants in equivariant birational geometry and their applications.

Margarida Melo (University of Roma Tre)
2021-04-29 12:00 - 13:00 | https://researchseminars.org/talk/ZAG/116/
Title: On the top weight cohomology of the moduli space of abelian varieties (https://researchseminars.org/talk/ZAG/116/)
by Margarida Melo (University of Roma Tre) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The moduli space of abelian varieties Ag admits well behaved toroidal compactifications whose dual complex can be given a tropical interpretation. Therefore, one can use the techniques recently developed by Chan-Galatius-Payne in order to understand part of the topology of Ag via tropical geometry. In this talk, which is based in joint work with Madeleine Brandt, Juliette Bruce, Melody Chan, Gwyneth Moreland and Corey Wolfe, I will explain how to use this setup, and in particular computations in the perfect cone compactification of Ag, in order to describe its top weight cohomology for g up to 7.

Maria Pe Pereira (Universidad Complutense de Madrid)
2021-05-04 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/117/
Title: Moderately Discontinuous Algebraic Topology (https://researchseminars.org/talk/ZAG/117/)
by Maria Pe Pereira (Universidad Complutense de Madrid) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
An algebraic or complex analytic subset in C^n has 2 natural metrics: the outer metric (restriction of the euclidean metric) and the inner metric (natural extension of the riemannian metric on the smooth part). These metrics considered up to bilipschitz mappings are analytic invariants, that is, they do not depend on the complex analytic embedding.
Recently there is an intense activity in bilipschitz geometry of germs and degenerations, enriching and providing finer information on
problems that were studied previously from the topological viewpoint (for multiplicity invariance of the germ or certain equisingularity notions).
In the works [1] and [2] we develop a new metric algebraic topology, called the Moderately Discontinuous Homology and Homotopy, in the context of subanalytic germs in R^n (with a supplementary metric structure) and more generally of (degenerating) subanalytic families. This theory captures bilipschitz information, or in other words, quasi isometric invariants, and aims to codify, in an algebraic way, part of the bilipschitz geometry.
A subanalytic germ is topologically a cone over its link and the moderately discontinuous theory captures the different speeds, with respect to the distance to the origin, in which the topology of the link collapses towards the origin. Similarly, in a degenerating subanalytic family, it captures the different speeds of collapsing with respect to the family parameter.
The MD algebraic topology satisfies all the analogues of the usual theorems in Algebraic Topology: long exact sequences for the relative case, Mayer Vietoris and Seifert van Kampen for special coverings...
In this talk, I will present the most important concepts in the theory and some results or applications that we got until the present.
[1] (with J. Fernandez de Bobadilla, S. Heinze, E. Sampaio) Moderately discontinuous homology. To appear in Comm. Pure App. Math.. Available in arXiv: 1910.12552
[2] (with J. Fernández de Bobadilla, S. Heinze) Moderately discontinuous homotopy. Submitted. Available in ArXiv:2007.01538

Mara Ungureanu (University of Freiburg)
2021-05-06 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/118/
Title: Counts of secant planes to varieties, Virasoro algebras, and universal polynomials (https://researchseminars.org/talk/ZAG/118/)
by Mara Ungureanu (University of Freiburg) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
For a curve in projective space, the varieties parametrising its secant planes are among the most studied objects in classical enumerative geometry. We shall start with an introduction to secant varieties and explain the role of degeneration arguments in understanding their geometry. We shall then explore the connection between the enumerative geometry of secant varieties and Virasoro algebras on one side, and tautological integrals on the other.

Stefan Kebekus (University of Freiburg)
2021-05-11 13:00 - 14:00 | https://researchseminars.org/talk/ZAG/119/
Title: Brauer-Manin obstruction on a simply connected fourfold and a Mordell theorem in the orbifold setting (https://researchseminars.org/talk/ZAG/119/)
by Stefan Kebekus (University of Freiburg) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups. In this paper, we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points. This is joint work with Jorge Pereira (IMPA) and Arne Smeets (Nijmegen)

Jack Rogers (University of Manchester)
2021-05-13 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/120/
Title: K-stability of smooth Fano SL2-threefolds (https://researchseminars.org/talk/ZAG/120/)
by Jack Rogers (University of Manchester) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
There has been much interest in K-stability since it was shown to be equivalent to the existence of Kähler-Einstein metrics by Chen-Donaldson-Sun. The theory of K-stability is now well developed, but practical methods to check whether a given variety is K-stable are hard to come by. Equivariant K-stability, introduced by Datar-Székelyhidi, makes finding such criteria easier for varieties with large automorphism groups.

If an algebraic group G acts on a variety X, the complexity of the action is the minimal codimension in X of the orbits of a Borel subgroup B of G (e.g. if T is a torus, the complexity zero T-varieties are the toric varieties). Conditions for K-stability have been found for toric varieties by Wang-Zhu, for complexity one T-varieties by Ilten-Süss and for all complexity zero varieties by Delcroix.

We will discuss the combinatorial description due to Timashev of complexity one G-varieties, and describe a practical method to check K-stability in the particular case of smooth Fano SL2-threefolds. In particular, this method proves the K-stability of several varieties not previously known to be K-stable, e.g. projective 3-space blown up along three disjoint lines. This is joint work with Hendrik Süss.

Jørgen Vold Rennemo (University of Oslo)
2021-05-18 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/121/
Title: K-theoretic sheaf counting invariants on C^4 (https://researchseminars.org/talk/ZAG/121/)
by Jørgen Vold Rennemo (University of Oslo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Oh and Thomas have recently defined a K-theoretic sheaf counting invariant for moduli spaces of sheaves on a Calabi-Yau 4-fold. One of the simplest examples of such a moduli scheme is the Hilbert scheme of n points on C^4. The topic of this talk is a proof of a formula for the generating functions of invariants of these Hilbert schemes, confirming a conjecture of Nekrasov (as well a generalisation to Quot schemes of C^4, conjectured by Nekrasov and Piazzalunga). This is joint work with Martijn Kool.

Marta Pieropan (Utrecht University)
2021-05-20 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/122/
Title: Campana points on Fano varieties (https://researchseminars.org/talk/ZAG/122/)
by Marta Pieropan (Utrecht University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We call Campana points an arithmetic notion of points on Campana's orbifolds that has been first studied by Campana and Abramovich, and that interpolates between the notions of rational and integral points. In this talk we introduce Campana points and a Manin type conjecture for Campana points on Fano varieties, and we present results for equivariant compactifications of vector groups (joint work with A. Smeets, S. Tanimoto, T. Várilly-Alvarado) and for toric varieties (joint work with D. Schindler).

Brendan Hassett (Brown University)
2021-05-25 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/123/
Title: Rationality of even-dimensional intersections of two real quadrics (https://researchseminars.org/talk/ZAG/123/)
by Brendan Hassett (Brown University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We consider rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four and discuss open cases in higher dimensions. (joint with János Kollár and Yuri Tschinkel)

Anthony Varilly-Alvarado (Rice University)
2021-05-27 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/124/
Title: Quasi-hyperbolicity via explicit symmetric differentials (https://researchseminars.org/talk/ZAG/124/)
by Anthony Varilly-Alvarado (Rice University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
A surface X is algebraically quasi-hyperbolic if it contains finitely many curves of genus 0 or 1. In 2006, Bogomolov and de Oliveira used asymptotic computations to show that sufficiently nodal surfaces of high degree in projective three-space carry symmetric differentials, and they used this to prove quasi-hyperbolicity of these surfaces. We explain how a granular analysis of their ideas, combined with computational tools and insights, yield explicit results for the existence of symmetric differentials, and we show how these results can be used to give constraints on the locus of rational curves on surfaces like the Barth Decic, Buechi's surface, and certain complete intersections of general type, including the surface parametrizing perfect cuboids, and the surface parametrizing magic squares of squares. This is joint work with Nils Bruin and Jordan Thomas.

Livia Campo (University of Birmingham)
2021-06-01 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/125/
Title: Fano 3-folds and double covers (https://researchseminars.org/talk/ZAG/125/)
by Livia Campo (University of Birmingham) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will propose a method to explicitly construct 52 deformation families of terminal Q-Fano 3-folds in codimension 4 and Fano index 2. Their structure descends from suitably crafted double covers. If time allows, I will briefly discuss the non-solidity of some of such families (in a joint work in progress with T. Guerreiro).

Enrico Fatighenti (Sapienza - Università di Roma)
2021-06-03 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/126/
Title: Fano varieties from homogeneous vector bundles (https://researchseminars.org/talk/ZAG/126/)
by Enrico Fatighenti (Sapienza - Università di Roma) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The idea of classifying Fano varieties using homogeneous vector bundles was behind Mukai's classification of prime Fano 3-folds. In this talk, we give a survey of some recent progress along the same lines, including a biregular rework of the non-prime Mori-Mukai 3-folds classification and some examples of higher-dimensional Fano varieties with special Hodge-theoretical properties.

Lev Borisov (Rutgers University)
2021-06-08 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/127/
Title: Explicit equations of surfaces of general type (https://researchseminars.org/talk/ZAG/127/)
by Lev Borisov (Rutgers University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will talk about progress in finding equations of special surfaces of general type, notably fake projective planes. This work was done over the last several years, in a series of joint papers with JongHae Keum, Sai Kee Yeung, Enrico Fatighenti and Anders Buch.

Fumiaki Suzuki (UCLA Mathematics)
2021-06-10 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/128/
Title: An O-acyclic variety of even index (https://researchseminars.org/talk/ZAG/128/)
by Fumiaki Suzuki (UCLA Mathematics) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will construct a family of Enriques surfaces parametrized by P^1 such that any multi-section has even degree over the base P^1. Over the function field of a complex curve, this gives the first example of an O-acyclic variety (H^i(X,O)=0 for i>0) whose index is not equal to one, and an affirmative answer to a question of Colliot-Thélène and Voisin. I will also discuss applications to related problems, including the integral Hodge conjecture and Murre’s question on universality of the Abel-Jacobi maps. This is joint work with John Christian Ottem.

Eveline Legendre (Institut de Mathématiques de Toulouse)
2021-06-15 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/129/
Title: Valuative stability for polarised varieties (https://researchseminars.org/talk/ZAG/129/)
by Eveline Legendre (Institut de Mathématiques de Toulouse) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will talk about a recent joint work with Ruadhai Dervan where we introduced a notion of valuative stability for polarised variety. This extends Fujita's valuative stability of Fano varieties. We show that valuative stability is equivalent to K-stability with respect to test configurations with integral central fibre.

Paolo Cascini (Imperial College London)
2021-06-17 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/130/
Title: Birational geometry of foliations on a complex threefold (https://researchseminars.org/talk/ZAG/130/)
by Paolo Cascini (Imperial College London) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Many results in the classical minimal model program, such as the existence of flips and the base point free theorem, admit a natural generalisation to the category of foliations defined over a complex threefold. Other results, instead, seem to behave differently, such as existence of flops and canonical models. I will survey about some of the recent progress in this direction. Joint work with C. Spicer.

Krishna Hanumanthu (Chennai Mathematical Institute)
2021-06-22 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/131/
Title: Rationality questions on Seshadri constants. (https://researchseminars.org/talk/ZAG/131/)
by Krishna Hanumanthu (Chennai Mathematical Institute) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Let X be a projective variety and let L be an ample line bundle on X. For a point x in X, the Seshadri constant of L at x is the infimum, over all curves C passing through x, of the ratios (L.C)/m, where (L.C) denotes the intersection product of L and C and m is the multiplicity of C at x. These constants were defined by J.-P. Demailly in 1990 and they shed light on the local behaviour of L at x and even say something about the nature of L and X. An important question about Seshadri constants is whether they can be irrational. They are expected to be irrational often, even though currently no examples are known. In this talk, we will focus on rational surfaces. We will discuss certain conjectures on linear systems of plane curves and show that Seshadri constants of some ample line bundles are irrational if these conjectures are true. This talk is based on joint works with B. Harbourne, \L. Farnik, J. Huizenga, D. Schmitz and T. Szemberg.

Howard Nuer (Technion, Israel Institute of Technology)
2021-06-24 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/132/
Title: The cohomology of the general stable sheaf on a K3 surface (https://researchseminars.org/talk/ZAG/132/)
by Howard Nuer (Technion, Israel Institute of Technology) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Let X be a K3 surface of Picard rank one and degree 2n with ample generator H. Let M_H(v) be the moduli space of Gieseker semistable sheaves on X with Mukai vector v. In this talk, we consider the weak Brill-Noether property for v, namely that the general sheaf in M_H(v) has at most one nonzero cohomology group. We show that given any positive rank r, there are only finitely many Mukai vectors of rank r failing weak Brill-Noether over all K3 surfaces of Picard rank one. We discuss our algorithm for finding the potential counterexamples and demonstrate the utility of our approach by discussing how we were able to classify all such counterexamples up to rank 20 and calculate the cohomology of the general sheaf in each case. Moreover, for fixed rank r, we give sharp bounds on n, d, and a that guarantee that a Mukai vector v=(r,dH,a) satisfies weak Brill-Noether. As a corollary, we provide another proof of the classification of Ulrich bundles on K3 surfaces of Picard rank one. In addition, we discuss the question of when the general sheaf in M_H(v) is globally generated. This joint work with Izzet Coskun and Kota Yoshioka makes crucial use of Bridgeland stability conditions and wall-crossing.

Javier Fernandez de Bobadilla (Basque Center for Applied Mathematics)
2021-06-29 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/133/
Title: The Brasselet-Schurmann-Yokura conjecture for L-classes on singular varieties (https://researchseminars.org/talk/ZAG/133/)
by Javier Fernandez de Bobadilla (Basque Center for Applied Mathematics) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The Brasselet-Schurmann-Yokura conjecture predicts the equality between the Hodge L-class and the Goresky-MacPherson L-class for compact complex algebraic varieties that are rational homology manifolds. In this talk, we give two different proofs of this conjecture. The first proof is for projective varieties, and it is based on cubical hyperresolutions, the Decomposition Theorem, and classical Hodge theory. This is a joint work with I. Pallares. The second proof is for general compact algebraic varieties by using the theory of mixed Hodge modules. This is a joint work with I. Pallares and M. Saito.

Sheng Meng (Korea Institute for Advanced Study)
2021-07-01 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/134/
Title: Automorphism group and its Jordan property (https://researchseminars.org/talk/ZAG/134/)
by Sheng Meng (Korea Institute for Advanced Study) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
A group is said to have Jordan property if there exists a Jordan constant J such that any finite subgroup has an abelian subgroup with index bounded by J. I will survey known results for Jordan property on certain automorphism groups and birational automorphism groups of varieties. I will also explain our recent progress on automorphism groups of compact spaces in Fujiki’s class C. This is a joint work with Fabio Perroni and De-Qi Zhang.

Taku Suzuki (Utsunomiya University)
2021-07-06 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/135/
Title: Higher order minimal families of rational curves on Fano manifolds (https://researchseminars.org/talk/ZAG/135/)
by Taku Suzuki (Utsunomiya University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The purpose of my talk is to introduce the notion of higher order minimal families of rational curves on Fano manifolds and to explain two results concerning it. One is that Fano manifolds are covered by higher rational manifolds if their Chern characters satisfy some positivity conditions. The other is a classification of embedded Fano manifolds covered by higher linear spaces.

Aline Zanardini (Leiden University)
2021-07-08 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/136/
Title: Stability of pencils of plane curves (https://researchseminars.org/talk/ZAG/136/)
by Aline Zanardini (Leiden University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk I will discuss some recent results on the problem of classifying pencils of plane curves up to projective equivalence. We will see how the stability of a pencil is related to the stability of its generators, to the log canonical threshold, and to the multiplicities of a base point. In particular, I will present complete stability criteria for certain pencils of plane sextics called Halphen pencils of index two.

Zheng Zhang (ShanghaiTech)
2021-07-13 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/137/
Title: The moduli space of cubic surface pairs via the intermediate Jacobians of Eckardt cubic threefolds (https://researchseminars.org/talk/ZAG/137/)
by Zheng Zhang (ShanghaiTech) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We study the moduli space of pairs consisting of a smooth cubic surface and a transverse plane via a period map. More specifically, the construction associates to a cubic surface pair a so-called Eckardt cubic threefold which admits an involution, and the period map sends the pair to the anti-invariant part of the intermediate Jacobian. Our main result is that the global Torelli theorem holds for the period map (in other words, the period map is injective). The key ingredients of the proof include a description of the anti-invariant part of the intermediate Jacobian as a Prym variety of a branched cover and a detailed study of certain positive dimensional fibers of the corresponding Prym map. This is joint work with S. Casalaina-Martin.

Nicole Lemire (University of Western Ontario)
2021-07-15 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/138/
Title: Codimension 2 cycles of classifying spaces of low-dimensional algebraic tori (https://researchseminars.org/talk/ZAG/138/)
by Nicole Lemire (University of Western Ontario) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Let T be an algebraic torus over a field F, and let CH^2(BT) be the Chow group of codimension 2 cycles in its classifying space. Following work of Blinstein and Merkurjev on the structure of the torsion part of CH^2(BT), Scavia, in a recent preprint, found an example of an algebraic torus with non-trivial torsion in CH^2(BT). In joint work with Alexander Neshitov, we show computationally that the group CH^2(BT) is torsion-free for all algebraic tori of dimension at most 5 and determine the conjugacy classes of finite subgroups of GL_6(Z) which correspond to 6-dimensional tori with nontrivial torsion in CH^2(BT). Some interesting properties of the structure of low-dimensional algebraic tori are involved.

Eiji Inoue (RIKEN iTHEMS)
2021-07-20 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/139/
Title: Perelman's entropy and optimal degeneration (https://researchseminars.org/talk/ZAG/139/)
by Eiji Inoue (RIKEN iTHEMS) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Algebraic optimal degeneration of Fano variety along Kahler-Ricci flow was originally constructed by Chen-Sun-Wang and was deepened by Dervan-Szekelyhidi, Han-Li and recent Blum-Liu-Xu-Zhuang. The degeneration is a substantial intermediate for studying a Fano variety with Kahler-Ricci soliton appearing in the Gromov-Hausdorff limit of Kahler-Ricci flow. The degeneration is characterized by a valuation which maximizes `H-entropy' among all valuations. Motivated by these works, I would like to explain my ongoing attempt to optimal degeneration of polarized variety with respect to `mu-entropy'. The mu-entropy appears in my study on mu-cscK metrics and muK-stability, which I introduced to understand cscK metrics and Kahler-Ricci soliton in a unified way. Going deep into the story, we encounter Perelman's entropy, which turns out to be the origin of our story.

Swarnava Mukhopadhyay (Tata Institute of Fundamental Research)
2021-07-22 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/140/
Title: Hitchin connection for parabolic bundles (https://researchseminars.org/talk/ZAG/140/)
by Swarnava Mukhopadhyay (Tata Institute of Fundamental Research) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In a fundamental paper in1990, Hitchin considered the space of non-abelian theta functions/conformal blocks/Verlinde spaces from the view point of geometric-quantization (Konstant-Kirillov-Soureau) for the moduli space of principal bundles on a smooth projective curve. Hitchin found a flat projective connection that can be interpreted as identification of these spaces via a parallel transport along a path joining different curves in the Teichmuller space. In this talk, we will discuss a generalization of Hitchin's construction to the parabolic set-up. Namely we consider punctured curves and the moduli space of parabolic $G$ bundles and produce a flat projective connection that identifies sections of parabolic determinant bundles as the puncture curve varies. This is a joint work with Indranil Biswas and Richard Wentworth.

Kenneth Ascher (Princeton University)
2021-07-27 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/141/
Title: K-stability and birational geometry of moduli spaces of quartic K3 surfaces (https://researchseminars.org/talk/ZAG/141/)
by Kenneth Ascher (Princeton University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We discuss various compactifications of moduli spaces of quartic K3 surfaces constructed using GIT, hodge theory, and K-stability. This is based on joint work with Kristin DeVleming and Yuchen Liu.

Anand Sawant (Tata Institute of Fundamental Research)
2021-07-29 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/142/
Title: Near-rationality properties of norm varieties (https://researchseminars.org/talk/ZAG/142/)
by Anand Sawant (Tata Institute of Fundamental Research) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The standard norm varieties played a crucial role in Voevodsky's proof of the Bloch-Kato conjecture. I will discuss various near-rationality concepts for smooth projective varieties and describe known near-rationality results for standard norm varieties. I will then outline an argument showing that a standard norm variety over a field of characteristic 0 is universally R-trivial after passing to the algebraic closure of the base field. The talk is based on joint work with Chetan Balwe and Amit Hogadi.

Andrea Fanelli (Institut de Mathématiques de Bordeaux)
2021-09-02 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/143/
Title: Rational simple connectedness and Fano threefolds (https://researchseminars.org/talk/ZAG/143/)
by Andrea Fanelli (Institut de Mathématiques de Bordeaux) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The notion of rational simple connectedness can be seen as an algebro-geometric analogue of simple connectedness in topology. The work of de Jong, He and Starr has already produced several recent studies to understand this notion.
In this talk I will discuss the joint project with Laurent Gruson and Nicolas Perrin to study rational simple connectedness for Fano threefolds via explicit methods from birational geometry.

Christian Schnell (Stony Brook University)
2021-09-07 09:00 - 10:00 | https://researchseminars.org/talk/ZAG/144/
Title: A new finiteness theorem for variations of Hodge structure (https://researchseminars.org/talk/ZAG/144/)
by Christian Schnell (Stony Brook University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will talk about a new finiteness theorem for variations of Hodge structure. It is a generalization of the Cattani-Deligne-Kaplan theorem from Hodge classes to so-called self-dual (and anti-self-dual) classes. For example, among integral cohomology classes of degree 4, those of type (4,0) + (2,2) + (0,4) are self-dual, and those of type (3,1) + (1,3) are anti-self-dual. The result is suggested by considerations in theoretical physics, and the proof uses o-minimality and the definability of period mappings. This is joint work with Benjamin Bakker, Thomas Grimm, and Jacob Tsimerman.

Stefano Filipazzi (École polytechnique fédérale de Lausanne)
2021-09-09 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/145/
Title: On the boundedness of elliptically fibered varieties (https://researchseminars.org/talk/ZAG/145/)
by Stefano Filipazzi (École polytechnique fédérale de Lausanne) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk, we will survey some ideas to address the boundedness of varieties admitting an elliptic fibration. After introducing general ideas, we will discuss how they apply concretely to special classes of varieties: n-folds of Kodaira dimension n-1, and elliptic Calabi--Yau varieties. Part of this talk is based on current work in progress joint with C.D. Hacon and R. Svaldi.

Jaroslaw Wisniewski (Institute of Mathematics, University of Warsaw)
2021-09-14 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/146/
Title: Rational maps via C* actions (https://researchseminars.org/talk/ZAG/146/)
by Jaroslaw Wisniewski (Institute of Mathematics, University of Warsaw) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will recall old results and present new facts linking varieties with C^* action to birational maps. I particular I will talk about recent results obtained together with Michalek, Monin, Occhetta, Romano, and Sola Conde.

Geordie Williamson (Sydney Mathematical Research Institute)
2021-09-16 21:00 - 22:00 | https://researchseminars.org/talk/ZAG/147/
Title: Geometric extensions (https://researchseminars.org/talk/ZAG/147/)
by Geordie Williamson (Sydney Mathematical Research Institute) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The Decomposition Theorem is probably my favourite theorem. I'll start by recalling the statement and trying to explain a little of why it is so remarkable. One consequence is that one could imagine an alternate history of the subject, in which intersection cohomology complexes were discovered without knowing about perverse sheaves. I'll explain how this might have occurred, and how it led Soergel and Juteau-Mautner-Williamson to the study of parity sheaves. Parity sheaves need strong parity vanishing assumptions, which somewhat restrict their utility (particularly for applications outside of geometric representation theory). The final goal of my talk is to explain a recent theorem (joint with Chris Hone) which proves the existence and uniqueness of "geometric extensions" in broad generality. The upshot is that parity sheaves probably don't need parity after all.

Alexander Perry (University of Michigan)
2021-09-21 13:00 - 14:00 | https://researchseminars.org/talk/ZAG/148/
Title: Serre functors of semiorthogonal components (https://researchseminars.org/talk/ZAG/148/)
by Alexander Perry (University of Michigan) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The Serre functor of a triangulated category is one of its most important invariants, playing the role of the dualizing complex of a variety in noncommutative algebraic geometry. I will explain how to describe the Serre functors of many semiorthogonal components of varieties in terms of spherical twists. In the case of Kuznetsov components of Fano complete intersections, this leads to a proof of a conjecture of Katzarkov and Kontsevich on the dimensions of such categories, and implies the nonexistence of Serre invariant stability conditions when the degrees of the complete intersection do not all coincide. This is joint work with Alexander Kuznetsov.

June Huh (Princeton University)
2021-09-23 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/149/
Title: Lorentzian polynomials (https://researchseminars.org/talk/ZAG/149/)
by June Huh (Princeton University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions. Specific examples can be constructed from graphs, convex bodies, stable polynomials, irreducible representations of general linear groups, and in particular, projective varieties. I will explain the main ideas from the viewpoint of algebraic geometry. Based on joint work with Petter Branden.

Giulia Sacca (Columbia University)
2021-09-28 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/150/
Title: Compactification of Lagrangian fibrations (https://researchseminars.org/talk/ZAG/150/)
by Giulia Sacca (Columbia University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Lagrangian fibered Hyper-Kahler manifolds are the natural generalization of elliptic K3 surfaces and have been used to study and construct examples of compact Hyper-Kahler manifolds and (possibly singular) symplectic varieties. In this talk I will talk about some recent compactification techniques I introduced for quasi-projective Lagrangian fibrations, with applications to the study of Prym, Intermediate Jacobian, and Albanese fibrations.

Amnon Neeman (Australian National University)
2021-09-30 21:00 - 22:00 | https://researchseminars.org/talk/ZAG/151/
Title: Uniqueness theorems for dg enhancements (https://researchseminars.org/talk/ZAG/151/)
by Amnon Neeman (Australian National University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Abstract: The study of Fourier-Mukai transforms has led to many fascinating developments, and in this talk we will focus on just one of them. In a 2004 article Bondal, Larsen and Lunts made the daring conjecture that geometric categories should have unique enhancements, and this remarkable conjecture turns out to be true. We will describe the series of theorems that have come out of this conjecture. The recent work, which we will get to at the end of the talk, is joint with Alberto Canonaco and Paolo Stellari.

Ana Caraiani (Imperial College London)
2021-10-05 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/152/
Title: On the cohomology of Hilbert modular varieties with torsion coefficients (https://researchseminars.org/talk/ZAG/152/)
by Ana Caraiani (Imperial College London) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Shimura varieties are certain moduli spaces equipped with many symmetries, that play an important role in the Langlands programme. For example, Hilbert modular varieties are quotients of the product of several copies of the upper-half plane by certain arithmetic groups. I will discuss a general conjecture on the cohomology of Shimura varieties with torsion coefficients, which states that the non-degenerate part of their cohomology is concentrated in the middle degree. I will give an overview of an approach to this conjecture introduced in joint work with Peter Scholze. This approach relies on the geometry of the Hodge-Tate period morphism, which I will describe, and on certain technical computations. I will then specialise to the case of Hilbert modular varieties and explain a modified version of this approach that relies on an instance of geometric Jacquet-Langlands functoriality for the fibers of the Hodge-Tate period morphism. This is joint work with Matteo Tamiozzo.

Yongnam Lee (Korea Advanced Institute of Science and Technology)
2021-10-07 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/153/
Title: Dominant rational maps from a very general hypersurface (https://researchseminars.org/talk/ZAG/153/)
by Yongnam Lee (Korea Advanced Institute of Science and Technology) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
There has been recent interest in studying measures of irrationality for hypersurfaces $X$ of high degree in $\mathbb P^{n+1}$. The degree of irrationality of $X$ is defined as the minimal degree of dominant rational maps from $X$ to $\mathbb P^n$. It is known that the degree of irrationality of $X$ is $d-1$ if $X$ is a very general hypersurface of degree $d\ge 2n+1$. In this talk, from a different point of view we will discuss dominant rational maps of finite degree from a very general hypersurface $X$ of degree $d\ge n+3$ in $\mathbb P^{n+1}$ to any smooth projective variety $Z$. The finite theorem states that these form a finite set, up to birational equivalence of $Z$, if $Z$ is a variety of general type. It is an interesting question to determine $Z$ when $Z$ is not birational to $X$. It is conjecturally expected that $Z$ is rationally connected if $X$ is a very general hypersurface of degree $d\ge n+3$. In this talk, I will present some partial results for this expectation. This talk combines the joint work with Gian Pietro Pirola and the joint work with De-Qi Zhang.

Richard Thomas (Imperial College London)
2021-10-12 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/154/
Title: Higher rank DT theory from curve counting (https://researchseminars.org/talk/ZAG/154/)
by Richard Thomas (Imperial College London) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X. Along the way we also show they are determined by rank 0 invariants counting sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.

Fei Hu (University of Oslo)
2021-10-14 12:00 - 13:00 | https://researchseminars.org/talk/ZAG/155/
Title: A dynamical approach to generalized Weil's Riemann hypothesis (https://researchseminars.org/talk/ZAG/155/)
by Fei Hu (University of Oslo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Inspired by a result of Esnault and Srinivas on automorphisms of surfaces and recent advances in complex dynamics, Truong raised a question on the comparison of two dynamical degrees, which are defined using pullback actions of dynamical correspondences on numerical cycle class groups and cohomology groups, respectively. An affirmative answer to his question would surprisingly imply Weil’s Riemann hypothesis. In this talk, I shall discuss some partial results in the cases of Abelian varieties and surfaces. This is based on joint work with Tuyen Truong.

Ronan Terpereau (Université de Bourgogne)
2021-10-19 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/156/
Title: Real structures on almost homogeneous varieties (https://researchseminars.org/talk/ZAG/156/)
by Ronan Terpereau (Université de Bourgogne) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk, we will be interested in the real structures on the almost homogeneous varieties; these are the complex algebraic varieties X endowed with an action of a reductive algebraic group G such that G acts on X with a dense open orbit. We will see how to determine when a real structure on X exists and, if so, how to describe and count all of them. In particular, we will try to illustrate our approach on two classical families of almost homogeneous varieties: the horospherical varieties (which include toric manifolds and flag varieties) and the SL(2)-almost homogeneous threefolds (which include the Fano threefolds P3, Q3, V5, and V22). This is a joint work with Lucy Moser-Jauslin (IMB, Dijon).

Donu Arapura (Purdue University)
2021-10-21 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/157/
Title: Euler characteristics of aspherical Kähler manifolds (https://researchseminars.org/talk/ZAG/157/)
by Donu Arapura (Purdue University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
This is joint work with Botong Wang. I want to start by recalling the Hopf-Singer conjecture on the sign of the Euler characteristic of a compact aspherical manifold. In the K\”ahler setting, we have a stronger conjecture involving Euler characteristics of perverse sheaves. Our results are that the stronger conjecture holds when X is compact K\”ahler with nonpositive curvature, or X is aspherical projective with a faithful rigid local system. I will sketch the proofs, and describe with some illustrative examples.

Fenglong You (University of Oslo)
2021-10-26 13:00 - 14:00 | https://researchseminars.org/talk/ZAG/158/
Title: Degenerations, fibrations and mirror symmetry (https://researchseminars.org/talk/ZAG/158/)
by Fenglong You (University of Oslo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Given a Tyurin degeneration of a Calabi-Yau variety to a union of two quasi-Fano varieties, Doran-Harder-Thompson conjectured that the Landau-Ginzburg models of two quasi-Fano varieties can be glued to the mirror Calabi-Yau variety which admits a P^1-fibration. I will talk about a generalization of this conjecture to degenerations of quasi-Fano varieties. I will explain some evidence of this conjecture including a gluing formula for (relative) periods.

Viatcheslav Kharlamov (University of Strasbourg)
2021-10-28 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/159/
Title: New examples of surgery invariant counts in real algebraic geometry (https://researchseminars.org/talk/ZAG/159/)
by Viatcheslav Kharlamov (University of Strasbourg) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Initial Welschinger invariants, as well as their various generalizations, are very sensitive, in general , to a change of topology of the underlying real structure. However, it was soon noticed that some natural combinations of them have a stronger invariance property (remaining also non-trivial in many interesting cases), the property that I call "surgery invariance": for a given complex deformation class of a variety, they no more depend on a chosen real structure. The starting example is the signed count of real lines on cubic surfaces in accordance with B.~Segre's division of such lines in 2 kinds, hyperbolic and elliptic. It is this example that originated a discovery of similar counts on higher dimensional hypersurfaces and complete intersections, and served as one of the impulses for a development of an integer valued real Schubert calculus. In this talk (based on a work in progress, joint with Sergey Finashin) I intend to discuss extending of the above cubic surface example in a bit different direction: from lines on a cubic surface to lines, and even higher degree rational curves, on other del Pezzo surfaces.

Elana Kalashnikov (Harvard University)
2021-11-02 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/160/
Title: Undoing toric degenerations: an analogue of Greene-Plesser mirror symmetry for the Grassmannian (https://researchseminars.org/talk/ZAG/160/)
by Elana Kalashnikov (Harvard University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The most basic construction of mirror symmetry compares the Calabi–Yau hypersurfaces of projective space and projective space quotient a finite group G. There is a natural analogue of this finite group action on the Grassmannian Gr(n, r). In this talk, I'll explain how toric degenerations, blow-ups, variation of GIT and mirror symmetry relate the Calabi–Yau hypersurfaces of Gr(n,r) and Gr(n,r)/G. This is joint work with Tom Coates and Charles Doran.

François Loeser (Sorbonne Université)
2021-11-04 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/161/
Title: A motivic version of topological mirror symmetry (https://researchseminars.org/talk/ZAG/161/)
by François Loeser (Sorbonne Université) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Hausel and Thaddeus have conjectured that the moduli spaces of twisted SL_n- and PGL_n-Higgs bundles on a smooth projective curve have the same (twisted) stringy Hodge numbers. This was recently proven by Groechenig, Wyss and Ziegler using $p$-adic integration. In this talk we shall explain how we can prove using motivic integration that the result holds in the Grothendieck ring of rational Chow motives. This is joint work with D. Wyss.

Farbod Shokrieh (University of Washington)
2021-11-09 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/162/
Title: Heights and moments of abelian varieties (https://researchseminars.org/talk/ZAG/162/)
by Farbod Shokrieh (University of Washington) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We give a formula which, for a principally polarized abelian variety $(A, \lambda)$ over a number field (or a function field), relates the stable Faltings height of $A$ with the N\'eron--Tate height of a symmetric theta divisor on $A$. Our formula involves invariants arising from tropical geometry. We also discuss the case of Jacobians in some detail, where graphs and electrical networks will play a key role. (Based on joint works with Robin de Jong.)

Rosa Winter (King's College London)
2021-11-11 13:00 - 14:00 | https://researchseminars.org/talk/ZAG/163/
Title: Concurrent exceptional curves on del Pezzo surfaces of degree one and torsion points on elliptic fibrations (https://researchseminars.org/talk/ZAG/163/)
by Rosa Winter (King's College London) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Let S be a del Pezzo surface of degree one. Then S contains 240 exceptional curves over an algebraically closed field. After blowing up a specific point one obtains an elliptic surface E, where the exceptional curves correspond to 240 sections. At most 16 exceptional curves gan go through the same point on S, and when this happens, the corresponding point on E is torsion on its fiber. In this talk I will consider the question how many exceptional curves can go through a point on S for which the corresponding point on E is non-torsion on its fiber. First of all I will explain how this question came up when studying the density of the set of rational points on del Pezzo surfaces of degree one. I will then show that if at least 9 exceptional curves intersect in a point on~S, the corresponding point on E is torsion on its fiber. This is less trivial than one might think by looking at the Mordell--Weil rank of E. Finally, in joint work with Julie Desjardins we show that 7 exceptional curves can go through a non-torsion point, and the question if 8 exceptional curves can go through a non-torsion point is still work in progress. I will show how one might try to tackle this.

Mark Gross (University of Cambridge)
2021-11-16 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/164/
Title: Open FJRW theory (https://researchseminars.org/talk/ZAG/164/)
by Mark Gross (University of Cambridge) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will describe joint work with Tyler Kelly and Ran Tessler. FJRW (Fan-Jarvis-Ruan-Witten) theory is an enumerative theory of quasi-homogeneous singularities, or alternatively, of Landau-Ginzburg models. It associates to a potential W:C^n -> C given by a quasi-homogeneous polynomial moduli spaces of (orbi-)curves of some genus and marked points along with some extra structure, and these moduli spaces carry virtual fundamental classes as constructed by Fan-Jarvis-Ruan. Here we specialize to the case W=x^r+y^s and construct an analogous enumerative theory for disks. We show that these open invariants provide perturbations of the potential W in such a way that mirror symmetry becomes manifest. Further, these invariants are dependent on certain choices of boundary conditions, but satisfy a beautiful wall-crossing formalism.

Oscar Garcia-Prada (ICMAT)
2021-11-18 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/165/
Title: Higgs bundles and higher Teichmüller spaces (https://researchseminars.org/talk/ZAG/165/)
by Oscar Garcia-Prada (ICMAT) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
It is well-known that the Teichmüller space of a compact real surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2,R). Higher Teichmüller spaces are generalizations of this, where PSL(2,R) is replaced by certain simple non-compact real Lie groups of higher rank. As for the usual Teichmüller space, these spaces consist entirely of discrete and faithful representations. Several cases have been identified over the years. First, the Hitchin components for split groups, then the maximal Toledo invariant components for Hermitian groups, and more recently certain components for SO(p,q). In this talk, I will describe a general construction of all possible higher Teichmüller spaces, and a parametrization of them using the theory of Higgs bundles, given in joint work with Bradlow, Collier, Gothen and Oliveira.

Sung Rak Choi (Yonsei University)
2021-11-23 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/166/
Title: Subadditivity theorem for Okounkov bodies (https://researchseminars.org/talk/ZAG/166/)
by Sung Rak Choi (Yonsei University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We will investigate the subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As an application, we obtain some numerical variants of the Iitaka conjecture. As a byproduct, we also obtain a criterion of birational isotriviality in terms of Okounkov bodies when the general fiber is of general type. We expect that our results will provide a new approach toward the Iitaka conjecture. This is a joint work with Jinhyung Park.

Christian Urech (École polytechnique fédérale de Lausanne)
2021-11-25 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/167/
Title: Actions of Cremona groups on CAT(0) cube complexes (https://researchseminars.org/talk/ZAG/167/)
by Christian Urech (École polytechnique fédérale de Lausanne) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Recently, in geometric group theory, isometric actions of groups on CAT(0) cube complexes have turned out to be powerful tools to study a broad range of groups. In this talk, I will explain the construction of CAT(0) cube complexes on which groups of birational transformations act by isometries and explain how to use these actions to deduce new and old group theoretical and dynamical results about Cremona groups. This is joint work with Anne Lonjou.

Fatemeh Rezaee (Loughborough University)
2021-11-30 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/168/
Title: Wall-crossing pathologies in dimension three (https://researchseminars.org/talk/ZAG/168/)
by Fatemeh Rezaee (Loughborough University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Running wall-crossing (with respect to Bridgeland stability conditions), I will describe the geometry of associated moduli spaces to canonical genus four curves, including the moduli space of PT stable pairs and the Hilbert scheme. Along the way, I will introduce a new wall-crossing phenomenon that induces non-Q-factorial singularities; hence, it cannot be detected as an operation in the Minimal Model Program of the corresponding moduli space, unlike the case for many surfaces.

Brian Lehmann (Boston College)
2021-12-02 13:00 - 14:00 | https://researchseminars.org/talk/ZAG/169/
Title: Rational curves on del Pezzo surfaces in characteristic p (https://researchseminars.org/talk/ZAG/169/)
by Brian Lehmann (Boston College) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Testa classified the components of the moduli space of rational curves on a del Pezzo surface over an algebraically closed field of characteristic 0. In characteristic p, new pathologies can appear. I will explain how such pathologies are predicted by Geometric Manin's Conjecture and how this perspective leads to a description of "problematic" surfaces. When no pathologies occur, we can completely classify the components of the moduli space of rational curves. This is joint work with Roya Beheshti, Eric Riedl, and Sho Tanimoto.

Jun-Muk Hwang (Korea Institute for Advanced Study)
2021-12-07 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/170/
Title: Minimal rational curves and 1-flat irreducible G-structures (https://researchseminars.org/talk/ZAG/170/)
by Jun-Muk Hwang (Korea Institute for Advanced Study) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
1-flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. In a joint work with Qifeng Li, we study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification result of Merkulov and Schwachhoefer on irreducible affine holonomy, the problem is reduced to the case when the VMRT at a general point of the uniruled projective manifold is isomorphic to a subadjoint variety. In the latter situation, we prove a stronger result that, without the assumption of 1-flatness, the structure arising from VMRT is always locally flat.

Helge Ruddat (University of Hamburg)
2021-12-09 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/171/
Title: Fano manifolds from smoothing toroidal crossing varieties (https://researchseminars.org/talk/ZAG/171/)
by Helge Ruddat (University of Hamburg) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I am reporting on a joint work with Corti, Felten, Petracci where we put particular singular log structures on toroidal crossing spaces. We prove various properties of the resulting reflexive log differential forms from using local log smooth log crepant resolutions. The properties give the ingredients to conclude the Hodge to de Rham degeneration and the smoothability of the toroidal crossing Fano variety. All Fanos with very amply anticanonical divisor are obtained from so called "admissible" log singularties by this method. We believe that yet more general log singularities give all Fano manifolds with non-empty anticanonical class by similar methods.

Laura Schaposnik (University of Illinois at Chicago)
2021-12-14 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/172/
Title: The geometry of Generalized Hyperpolygons, Hitchin systems and other scientific advances (https://researchseminars.org/talk/ZAG/172/)
by Laura Schaposnik (University of Illinois at Chicago) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan. After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Having made the connection to Hitchin moduli spaces, we will consider different geometric methods developed within that set up to study geometric properties of integrable systems and their dualities. Finally, we shall conclude the talk mentioning other directions in which one can apply geometric insights to make scientific advances.

Matthew Ballard (University of South Carolina)
2021-12-16 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/173/
Title: Stratified Mukai flops revisited (https://researchseminars.org/talk/ZAG/173/)
by Matthew Ballard (University of South Carolina) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We will describe how constructing Fourier-Mukai kernels for stratified Mukai flops fits into a general framework. Joint with N. Chidambaram and D. Favero.

Linquan Ma (Purdue University)
2021-12-21 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/174/
Title: The cohomology table of coherent sheaves on singular projective varieties (https://researchseminars.org/talk/ZAG/174/)
by Linquan Ma (Purdue University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The cohomology table of a coherent sheaf on a projective variety is numerical data of the dimension of each cohomoogy group of each twist of the sheaf. Eisenbud--Schreyer give a description of the cone of cohomology table of vector bundles and coherent sheaves on projective spaces. This leads to their proof of the Boij--Soderberg theory which describes the cone spanned by the Betti tables of graded modules over polynomial rings. In this talk, we give some extensions of these results of Eisenbud--Schreyer to singular projective varieties and singular standard graded rings. Our central technique is to use a sequence of coherent sheaves that behave like an Ulrich sheave asymptotically. We call such sequence a lim Ulrich sequence of sheaves and we can prove their existence in positive characteristic. This talk is largely based on joint work in progress with Srikanth Iyengar and Mark Walker.

Andrei Okounkov (Columbia University)
2022-02-03 09:00 - 10:00 | https://researchseminars.org/talk/ZAG/175/
Title: The unramified Eisenstein spectrum (https://researchseminars.org/talk/ZAG/175/)
by Andrei Okounkov (Columbia University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will report on joint work with David Kazhdan, in which we determine the unramified Eisenstein spectrum for
all split reductive groups over a global field. Our methods are rooted in topology and algebraic geometry.
Specifically, we relate the spectral decomposition to a decomposition of $T^*(B \backslash G / B)$ in in a certain cobordism group, where $G$ is the Langlands dual group. This works equally well for fields of positive and zero characteristic.

Takumi Murayama (Princeton University)
2022-01-11 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/176/
Title: The Kawamata–Viehweg vanishing theorem for schemes (https://researchseminars.org/talk/ZAG/176/)
by Takumi Murayama (Princeton University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In 1953, Kodaira proved what is now called the Kodaira vanishing theorem, which states that if L is an ample divisor on a complex projective manifold X, then H^i(X,-L) = 0 for all i < dim(X). Since then, Kodaira's theorem and its generalizations for complex projective varieties – in particular, the Kawamata–Viehweg vanishing theorem and its relative version due to Kawamata–Matsuda–Matsuki – have become indispensable tools in algebraic geometry over fields of characteristic zero, in particular in birational geometry and the minimal model program. However, while the goal in the minimal model program is to study birational equivalences between projective varieties, recent progress in the minimal model program has shown that it would be very useful to have a version of the Kawamata–Viehweg vanishing theorem that holds for schemes that are not necessarily projective varieties. Recently, I proved the relative Kawamata–Viehweg vanishing theorem for schemes of equal characteristic zero. My results are optimal given known counterexamples to vanishing theorems in positive and mixed characteristic, and have many applications to both algebraic geometry and commutative algebra. In this talk, I will discuss my vanishing theorem and several of its applications.

Kris Shaw (University of Oslo)
2022-01-13 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/177/
Title: A tropical approach to the enriched count of bitangents to quartic curves (https://researchseminars.org/talk/ZAG/177/)
by Kris Shaw (University of Oslo) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Using A1 enumerative geometry Larson and Vogt have provided an enriched count of the 28 bitangents to a quartic curve. In this talk, I will explain how these enriched counts can be computed combinatorially using tropical geometry. I will also introduce an arithmetic analogue of Viro’s patchworking for real algebraic curves which, in some cases, retains enough data to recover the enriched counts. This talk is based on joint work with Hannah Markwig and Sam Payne.

Eunjeong Lee (Institute for Basic Science, Center for Geometry and Physics)
2022-01-18 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/178/
Title: On smooth toric Richardson varieties (https://researchseminars.org/talk/ZAG/178/)
by Eunjeong Lee (Institute for Basic Science, Center for Geometry and Physics) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Schubert varieties and Richardson varieties are some of the most interesting subvarieties of the full flag varieties. A maximal torus acts on the full flag variety and these subvarieties are stable under the action. Considering the restriction of the moment map on Richardson varieties, we obtain Bruhat interval polytopes. The combinatorics of Bruhat interval polytopes play an important role in studying toric Richardson varieties. In this talk, we consider an interesting family of smooth toric Richardson varieties each element of which is associated with a cubic Bruhat interval polytope, called a toric variety \textit{of Catalan type}. We study the relationship between the isomorphism classes of toric varieties of Catalan type and polygon triangulations. Moreover, we consider the isomorphism classes of toric Schubert varieties which may not be of Catalan type. This talk is based on joint work with Mikiya Masuda and Seonjeong Park.

Ehud Hrushovski (University of Oxford)
2022-01-20 11:30 - 12:30 | https://researchseminars.org/talk/ZAG/179/
Title: Old and new results in the model theory of finite fields (https://researchseminars.org/talk/ZAG/179/)
by Ehud Hrushovski (University of Oxford) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Ax's determination of the first-order theory of finite fields in 1968 was the starting point of a rich chapter of model theory; highlights include the Denef-Loeser connection between definable sets over pseudo-finite fields and motives, their use in motivic integration, the Chatzidakis-Van den Dries-Macintyre measure theory, the strong Szemeredi regularity lemma for definable sets and applications to expansion due to Terry Tao. I will sketch a path through some of this material, aiming to touch on recent work of Levi and Chevalier on higher regularity, and on open problems that arise when an additive character is added to the structure.

Hannah Larson (Stanford University)
2022-01-25 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/180/
Title: The rational Chow rings of M_7, M_8, and M_9 (https://researchseminars.org/talk/ZAG/180/)
by Hannah Larson (Stanford University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The rational Chow ring of the moduli space M_g of curves of genus g is known for g up to 6. In each of these cases, the Chow ring is tautological (generated by certain natural classes known as kappa classes). In joint work with Sam Canning, we prove that the rational Chow ring of M_g is tautological for g = 7, 8, 9, thereby determining the Chow rings by earlier work of Faber. In this talk, I will give an overview of our approach, with particular focus on the locus of tetragonal curves (special curves admitting a degree 4 map to P^1).

Sukmoon Huh (Sungkyunkwan Universityok)
2022-01-27 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/181/
Title: Torelli problem on logarithmic sheaves (https://researchseminars.org/talk/ZAG/181/)
by Sukmoon Huh (Sungkyunkwan Universityok) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The logarithmic sheaf associated to a reduced divisor, is the sheaf of differential 1-forms with logarithmic poles along the divisor, and it was introduced by P. Deligne to define a mixed Hodge structure on the complement of the divisor. There have been a great deal of study on this subject, and one of the questions is whether the sheaf determines the divisor or not, which we call the Torelli problem. In case of general hyperplane arrangements on projective spaces, I. Dolgachev and M. Kapranov gave a positive answer to the Torelli problem when the number of hyperplanes is big enough, and then later J. Valles gave a complete answer. In this talk we give several other results on the Torelli problem, and report our recent result, in which we introduce two different approachs to have a positive answer on the problem. This is a joint work with S. Marchesi, J. Pons-Llopis and J. Valles.

Soheyla Feyzbakhsh (Imperial College London)
2022-02-01 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/182/
Title: Moduli spaces of stable objects in the Kuznetsov component of cubic threefolds (https://researchseminars.org/talk/ZAG/182/)
by Soheyla Feyzbakhsh (Imperial College London) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We will first discuss a general criterion that ensures a fractional Calabi-Yau category of dimension less than or equal to 2 admits a unique Serre-invariant stability condition up to the action of the universal cover of GL+(2, R). This result can be applied to a certain triangulated subcategory (called the Kuznetsov component) of the bounded derived category of coherent sheaves on a cubic threefold. As an application, we will prove (i) a categorical version of the Torelli theorem holds for cubic threefolds, and (ii) the moduli space of Ulrich bundles of fixed rank r greater than or equal to 2 on a cubic threefold is irreducible. The talk is based on joint work with Laura Pertusi and a group project with A. Bayer, S.V. Beentjes, G. Hein, D. Martinelli, F. Rezaee and B. Schmidt.

Sean Keel (The University of Texas at Austin)
2022-02-08 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/183/
Title: Mirror symmetry and analytic disks (https://researchseminars.org/talk/ZAG/183/)
by Sean Keel (The University of Texas at Austin) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will explain my recent construction, joint with Tony Yu, of the mirror to an affine log CY manifold as the spectrum of an algebra with a canonical basis and structure constants naive counts of Berkovich analytic disks. I wont assume any previous knowledge of Berkovich geometry.

Mario Kummer (TU Dresden)
2022-02-10 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/184/
Title: Secant varieties of real curves (https://researchseminars.org/talk/ZAG/184/)
by Mario Kummer (TU Dresden) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
After an introduction to hyperbolic polynomials and their relevance in convex optimization, I will explain a construction of hyperbolic polynomials using secant varieties of curves. Further, I will relate this to the theory of Ulrich sheaves.

Alena Pirutka (Courant Institute of Mathematical Sciences, New York University)
2022-02-15 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/185/
Title: On local-global principles and Galois cohomology (https://researchseminars.org/talk/ZAG/185/)
by Alena Pirutka (Courant Institute of Mathematical Sciences, New York University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk we will review several questions of local-global type for schemes of dimension 3 in the arithmetic (curves over semi-global fields) and geometric (threefolds over a finite field) settings.

David Favero (University of Alberta)
2022-02-17 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/186/
Title: Path Homotopy Quiver Algebras and Mirror Symmetry (https://researchseminars.org/talk/ZAG/186/)
by David Favero (University of Alberta) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Given a quiver (directed graph) embedded in a topological space, one can consider the algebra of directed paths up to homotopy. Conversely, given a certain type of algebra, I will construct a quiver and a topological space which recovers this algebra by the above procedure. These constructions induce an equivalence between the derived category of the algebra and a certain subcategory of the derived category of sheaves on the topological space. As applications, I will recover some examples of homological mirror symmetry for Berglund-Hubsch-Krawitz mirrors and for toric varieties following work of Bondal and Fan-Lui-Treumann-Zaslow.

Alicia Dickenstein (Universidad de Buenos Aires)
2022-02-22 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/187/
Title: Iterated and mixed discriminants (https://researchseminars.org/talk/ZAG/187/)
by Alicia Dickenstein (Universidad de Buenos Aires) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Classical work by Salmon and Bromwich classified singular intersections of two quadric surfaces. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in the plane and used by Schafli for the study of hyperdeterminants. More recently, the problem has been revisited with similar tools in the context of geometric modeling and a generalization to the case of two higher dimensional quadric hypersurfaces was given by Ottaviani. In joint work with Sandra di Rocco and Ralph Morrison, we propose and study a generalization of this question for systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant. We define a related polynomial called the multivariate iterated discriminant. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre-Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant.

Diane Maclagan (University of Warwick)
2022-02-24 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/188/
Title: Toric and tropical Bertini theorems in arbitrary characteristic (https://researchseminars.org/talk/ZAG/188/)
by Diane Maclagan (University of Warwick) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The classical Bertini theorem on irreducibility when intersecting by hyperplanes is a standard part of the algebraic geometry toolkit. This was generalised recently, in characteristic zero, by Fuchs, Mantova, and Zannier to a toric Bertini theorem for subvarieties of an algebraic torus, with hyperplanes replaced by subtori. I will discuss joint work with Gandini, Hering, Mohammadi, Rajchgot, Wheeler, and Yu in which we give a different proof of this theorem that removes the characteristic assumption. An application is a tropical Bertini theorem.

Georg Oberdieck (Hausdorff Center for Mathematics)
2022-03-01 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/189/
Title: Holomorphic anomaly equations for the Hilbert schemes of points of a K3 surface (https://researchseminars.org/talk/ZAG/189/)
by Georg Oberdieck (Hausdorff Center for Mathematics) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Holomorphic anomaly equations are certain structural properties predicted by physics for the Gromov-Witten theory of Calabi-Yau manifolds. In this talk I will explain the conjectural form of these equations for the Hilbert scheme of points of a K3 surface, and explain how to prove them for genus 0 and up to three markings. As a corollary, for fixed n, the (reduced) quantum cohomology of Hilb^n K3 is determined up to finitely many coefficients.

Johannes Nordstrom (University of Bath)
2022-03-03 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/190/
Title: K3 surfaces and twisted connected sum G2-manifolds (https://researchseminars.org/talk/ZAG/190/)
by Johannes Nordstrom (University of Bath) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The twisted connected sum construction of Kovalev produces many examples of closed Riemannian 7-manifolds with holonomy group G_2 (a special class of Ricci-flat manifolds), starting from complex algebraic geometry data like Fano 3-folds. If the pieces admit automorphisms, then adding an extra twist to the construction yields examples with a wider variety of topological features. I will outline the constructions, and describe how a good understanding of moduli of K3 surfaces appearing as anticanonical divisors in Fano 3-folds is used in a crucial way to match the pieces in the gluing procedure. This is joint work with Diarmuid Crowley and Sebastian Goette.

Daniel Bragg (University of California, Berkeley)
2022-03-08 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/191/
Title: Compact supersingular twistor spaces (https://researchseminars.org/talk/ZAG/191/)
by Daniel Bragg (University of California, Berkeley) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Supersingular twistor spaces are certain families of K3 surfaces over A^1 associated to a supersingular K3 surface. We will describe a geometric construction that produces families of K3 surfaces over P^1 which compactify supersingular twistor spaces. The key input is a construction relating Brauer classes of order p on a scheme of characteristic p to certain sheaves of twisted differential operators. We will give some results on the geometry of compactified supersingular twistor spaces, and some applications.

Jean-Yves Welschinger (Institut Camille Jordan, Université Lyon 1)
2022-03-10 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/192/
Title: Shellable tilings on simplicial complexes (https://researchseminars.org/talk/ZAG/192/)
by Jean-Yves Welschinger (Institut Camille Jordan, Université Lyon 1) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will introduce a notion of shellable tilings on finite simplicial complexes, prove their existence after finitely many stellar subdivisions and discuss their relations with discrete Morse theory and h-vectors. Every shelled tiling computes the (co)homology of the complex via two spectral sequences. As a consequence, I deduce the existence of piecewise-linear pinched handle decompositions for all closed triangulated manifolds.

Guolei Zhong (Institute for Basic Science)
2022-03-15 09:00 - 10:00 | https://researchseminars.org/talk/ZAG/193/
Title: Compact Kähler threefolds with the action of an abelian group of maximal dynamical rank (https://researchseminars.org/talk/ZAG/193/)
by Guolei Zhong (Institute for Basic Science) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Let X be a compact Kaehler manifold. It is proved by Dinh and Sibony that, for any abelian subgroup G of the automorphism group Aut(X), if G is of positive entropy, then G is free abelian with rank no more than dim(X)-1. In the past decade, when X is projective, the extremal case rank(G)=dim(X)-1 (being maximal) has been intensively studied by Zhang. In this talk, we consider the case when X is a general compact Kaehler 3-fold and rank(G)=2. By running the G-equivariant log minimal model program, we show that such X is either rationally connected, or bimeromorphic to a quasi-etale quotient of a complex 3-torus.

Lawrence Jack Barrott (University of Leiden)
2022-03-17 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/194/
Title: Fibrations and degenerations of Calabi-Yau varieties via tropical geometry (https://researchseminars.org/talk/ZAG/194/)
by Lawrence Jack Barrott (University of Leiden) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
One expectation from the Doran-Harder-Thompson is that certain degenerations of Calabi-Yau's should correspond under mirror symmetry to fibrations by lower dimensional Calabi-Yau's. I will make this expectation precise, and explain how to prove it under certain assumptions using the Gross-Siebert program. The proof uses an old result of Deligne-Illusie together with modern deformation theory from Chan-Leung-Ma and Felten-Filip-Ruddat.

Tom Bridgeland (University of Sheffield)
2022-03-22 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/195/
Title: Geometric structures on spaces of quadratic differentials (https://researchseminars.org/talk/ZAG/195/)
by Tom Bridgeland (University of Sheffield) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
This talk is part of a long-running saga which aims to encode the Donaldson-Thomas invariants of a CY3 triangulated category in a geometric structure on its stability space. I will focus on a class of examples of such categories whose stability spaces are known to parameterise compact Riemann surfaces equipped with quadratic differentials. I will explain exactly what geometric structure we should look for on these spaces, and then use moduli spaces of Higgs bundles, flat connections, etc to construct it.

Masafumi Hattori (Kyoto University)
2022-03-24 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/196/
Title: On K-stability of Calabi-Yau fibrations (https://researchseminars.org/talk/ZAG/196/)
by Masafumi Hattori (Kyoto University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In K-stability, the characterization of K-stable varieties is well-studied when K_X is ample or X is a Calabi-Yau or Fano variety. However, K-stability of Fano fibrations or Calabi-Yau fibrations (i.e., K_X is relatively trivial) is not known much in algebraic geometry. On the other hand, cscK problems on fibrations are studied by Fine, Jian-Shi-Song and Dervan-Sektnan in Kahler geometry. We introduce adiabatic K-stability (If f:(X,H)\to (B,L) is a fibration of polarized varieties, this means that K-stability of (X,aH+L) for sufficiently small a) and show that adiabatic K-semistability of Calabi-Yau fibration implies log-twisted K-semistability of the base variety by applying the canonical bundle formula. If the base is a curve, we also obtain a partial converse. In this talk, I would like to explain our main results and their applications to rational elliptic surfaces and the conjecture of Miranda on Chow-stability of rational Weierstrass fibrations.

Tristan Collins (Massachusetts Institute of Technology)
2022-03-29 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/197/
Title: SYZ mirror symmetry for some Calabi-Yau surface pairs (https://researchseminars.org/talk/ZAG/197/)
by Tristan Collins (Massachusetts Institute of Technology) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will discuss a proof of a strong form of the SYZ mirror symmetry conjecture for Calabi-Yau surfaces pairs constructed from del Pezzo surfaces and rational elliptic surfaces. Time permitting, I will also mention some applications, including to the Torelli theorem for ALH* gravitational instantons. This is joint work with A. Jacob and Y.-S. Lin.

Julius Ross (University of Illinois Chicago)
2022-03-31 16:00 - 17:00 | https://researchseminars.org/talk/ZAG/198/
Title: Hodge-Riemann Classes and Schur Polynomials (https://researchseminars.org/talk/ZAG/198/)
by Julius Ross (University of Illinois Chicago) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The classical Hodge-Riemann bilinear relations are statements about the intersection form associated to the self-wedge product of a K\"ahler form on a compact complex manifold. Gromov initiated the question as to whether there are other cohomology that give rise these same bilinear relations, and proved that this is the case for the intersection of (possibly different) K\"ahler classes. In this talk I will discuss joint work with Matei Toma in which we prove that the Schur classes of ample vector bundles have the Hodge-Riemann bilinear relations (at least on $H^{1,1}$). This gives rise to a number of new inequalities among characteristic classes of ample vector bundles that should be thought of as generalizations of the Khovanskii-Tessier inequalities. And if time allows I will also discuss how this extends to the non-projective case and beyond.

Giulio Codogni (Università di Roma Tor Vergata)
2022-04-05 11:00 - 12:00 | https://researchseminars.org/talk/ZAG/199/
Title: Ample cone of KSB and K- moduli spaces (https://researchseminars.org/talk/ZAG/199/)
by Giulio Codogni (Università di Roma Tor Vergata) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will present some quantitative results about the ample cone of the moduli spaces of KSB stable and K-stable varieties of any dimension. These results follow from various higher dimensional generalizations of the Xiao-Cornalba-Harris slope inequality. Our proofs combine some new Noether and Castelnuovo inequalities with a careful study of the Harder-Narasimhan filtration of the push-forward of the log pluri-canonical bundles. The talk is based on a joint work with Luca Tasin and Filippo Viviani.

Antonio Laface (Universidad de Concepción)
2022-04-07 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/200/
Title: On effective cones of rational surfaces (https://researchseminars.org/talk/ZAG/200/)
by Antonio Laface (Universidad de Concepción) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Effective cones of rational surfaces have been intensively studied in the past few years, especially in the context of Nagata conjecture and Seshadri constants. After reviewing recent results on blow-ups of the projective plane at points in very general position, I will focus on blow-ups of toric surfaces at a general point, discussing examples of polyhedral and non-polyhedral effective cones. This is joint work with A-M. Castravet, J. Tevelev and L. Ugaglia.

Jose Ignacio Burgos Gil (Instituto de Ciencias Matemáticas)
2022-04-12 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/201/
Title: Chern-Weil theory and Hilbert-Samuel theorem for semi-positive singular toroidal metrics on line bundles (https://researchseminars.org/talk/ZAG/201/)
by Jose Ignacio Burgos Gil (Instituto de Ciencias Matemáticas) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk I will report on joint work with A. Botero, D. Holmes and R. de Jong. Using the theory of b-divisors and non-pluripolar products we show that Chen-Weil theory and a Hilbert Samuel theorem can be extended to a wide class of singular semi-positive metrics. We apply the techniques relating semipositive metrics on line bundles to b-divisors to study the line bundle of Siegel-Jacobi forms with the Peterson metric. On the one hand we prove that the ring of Siegel-Jacobi forms of constant positive relative index is never finitely generated, and we recover a formula of Tai giving the asymptotic growth of the dimension of the spaces of Siegel-Jacobi modular forms.

Max Lieblich (University of Washington)
2022-04-14 17:00 - 18:00 | https://researchseminars.org/talk/ZAG/202/
Title: Infinite rank vector bundles and applications (https://researchseminars.org/talk/ZAG/202/)
by Max Lieblich (University of Washington) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
This is a report on joint work with Johan de Jong and Minseon Shin. Grothendieck famously asked whether the Brauer group and cohomological Brauer group of a scheme coincide, a question which still lacks a good answer. Modern methods reduce this question to a question about the existence of non-zero vector bundles of finite rank on certain stacks. I will discuss what happens if one allows vector bundles of infinite rank, and how this infinite-rank version of the question is related to the resolution property for schemes. I will also discuss some basic questions about vector bundles of infinite rank on varieties, including the potential existence of some mysterious invariants.

Erik Paemurru (International Center for Mathematical Sciences – Sofia)
2022-04-19 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/203/
Title: Counting Divisorial Contractions with Centre a cAn-singularity (https://researchseminars.org/talk/ZAG/203/)
by Erik Paemurru (International Center for Mathematical Sciences – Sofia) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
First, we simplify the existing classification due to Kawakita and Yamamoto of 3-dimensional divisorial contractions with centre a cA_n-singularity. Next we consider divisorial contractions of discrepancy at least 2 to a fixed variety with centre a cA_n-singularity. We show that if there exists one such divisorial contraction, then there exist uncountably many such divisorial contractions.

Eva Elduque (Universidad Autónoma de Madrid)
2022-04-26 12:00 - 13:00 | https://researchseminars.org/talk/ZAG/204/
Title: Eigenspace decomposition of mixed Hodge structures on Alexander modules (https://researchseminars.org/talk/ZAG/204/)
by Eva Elduque (Universidad Autónoma de Madrid) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In previous work jointly with Geske, Herradón Cueto, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this talk, we will talk about the solution to this question, as well as some consequences and explicit computations. Joint work with Moises Herradón Cueto.

Theodoros Papazachariou (University of Essex)
2022-04-26 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/205/
Title: K-moduli for log Fano complete intersections (https://researchseminars.org/talk/ZAG/205/)
by Theodoros Papazachariou (University of Essex) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
An important category of geometric objects in algebraic geometry is smooth Fano varieties, which have positive curvature. These have been classified in 1, 10 and 105 families in dimensions 1, 2 and 3 respectively, while in higher dimensions the number of Fano families is yet unknown, although we know that their number is bounded. An important problem is compactifying these families into moduli spaces via K-stability. In this talk, I will describe the compactification of the family of Fano threefolds, which is obtained by blowing up the projective space along a complete intersection of two quadrics which is an elliptic curve, into a K-moduli space using Geometric Invariant Theory (GIT). A more interesting setting occurs in the case of pairs of varieties and a hyperplane section where the K-moduli compactifications tessellate depending on a parameter. In this case it has been shown recently that the K-moduli decompose into a wall-chamber decomposition depending on a parameter, but wall-crossing phenomena are still difficult to describe explicitly. Using GIT, I will describe an explicit example of wall-crossing in the K-moduli spaces, where both variety and divisor differ in the deformation families before and after the wall, given by log pairs of Fano surfaces of degree 4 and a hyperplane section.

Arman Sarikyan (University of Edinburgh)
2022-04-28 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/206/
Title: Equivariant pliability of the projective space (https://researchseminars.org/talk/ZAG/206/)
by Arman Sarikyan (University of Edinburgh) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
We classify finite subgroups $G\subset\mathrm{PGL}_4(\mathbb{C})$ such that $\mathbb{P}^3$ is not $G$-birational to conic bundles and del Pezzo fibrations, and explicitly describe all $G$-Mori fibre spaces that are $G$-birational to $\mathbb{P}^3$ for these subgroups. This is a joint work with I. Cheltsov

Kyoung-Seog Lee (University of Miami)
2022-05-03 15:00 - 16:00 | https://researchseminars.org/talk/ZAG/207/
Title: Derived categories and motives of moduli spaces of vector bundles on curves (https://researchseminars.org/talk/ZAG/207/)
by Kyoung-Seog Lee (University of Miami) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Derived categories and motives are important invariants of algebraic varieties invented by Grothendieck and his collaborators around the 1960s. In 2005, Orlov conjectured that they will be closely related and now there are several evidences supporting his conjecture. On the other hand, moduli spaces of vector bundles on curves provide attractive and important examples of algebraic varieties and there have been intensive works studying them. In this talk, I will discuss derived categories and motives of moduli spaces of vector bundles on curves and how they are related. Part of this talk is based on several joint works with I. Biswas, T. Gomez, H.-B. Moon and M. S. Narasimhan.

Ariyan Javanpeykar (Johannes Gutenberg University of Mainz)
2022-05-05 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/208/
Title: Shafarevich's conjecture for canonically polarized varieties revisited (https://researchseminars.org/talk/ZAG/208/)
by Ariyan Javanpeykar (Johannes Gutenberg University of Mainz) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Abstract: Arakelov-Parshin proved Shafarevich's conjecture for the (actual) moduli space M_g of curves of genus g (g>1). Namely, for every curve C, the set of isomorphism classes of non-isotrivial morphisms C \to M_g is finite. This finiteness result is similar to the theorem of De Franchis: for every hyperbolic curve X, the set of non-constant morphisms C->X is finite. Interestingly, M_g is hyperbolic (in any reasonable sense of the word "hyperbolic"). Is maybe the theorem of Arakelov-Parshin and De Franchis an instance of a more general finiteness property for hyperbolic varieties/stacks? The answer is no. The conclusion of the theorem of De Franchis fails for hyperbolic surfaces (such as X x X) and the conclusion of Arakelov-Parshin's theorem fails for the moduli space of canonically polarized surfaces (because it contains copies of X x X). How to remedy this? Guided by finiteness properties of compact hyperbolic varieties, we establish new finiteness properties for the moduli space CanPol of canonically polarized varieties by proving rigidity properties of pointed maps. Applications include bounds on dimensions of moduli spaces of maps to CanPol. Joint work with Steven Lu, Ruiran Sun, and Kang Zuo.

Emma Brakkee (University of Amsterdam)
2022-05-10 12:00 - 13:00 | https://researchseminars.org/talk/ZAG/209/
Title: General type results for moduli of hyperkahler varieties (https://researchseminars.org/talk/ZAG/209/)
by Emma Brakkee (University of Amsterdam) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In 2007, Gritsenko, Hulek and Sankaran proved that the moduli space of K3 surfaces of degree 2d is of general type when d>61. Their strategy is to reduce the question to the existence of a certain cusp form for an orthogonal modular variety. This method has been applied successfully to prove general type results for, among others, some moduli of higher-dimensional hyperkähler varieties. In this talk, we sketch the reduction argument and give general type results for some more types of hyperkähler moduli spaces. We also explain what the challenges are when trying to imitate the strategy for other moduli spaces of hyperkähler varieties. This is joint work in progress with I. Barros, P. Beri and L. Flapan.

Daniel Chan (UNSW Sydney)
2022-05-12 21:00 - 22:00 | https://researchseminars.org/talk/ZAG/210/
Title: The minimal model program for arithmetic surfaces enriched by a Brauer class (https://researchseminars.org/talk/ZAG/210/)
by Daniel Chan (UNSW Sydney) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Mori's minimal model program is a major organising principle for studying and classifying varieties $X$. It has been generalised in many directions, and in this talk, we examine a ``noncommutative'' one where $X$ is enriched by a Brauer class $\beta \in K(X)$. We focus on some new results where $X$ is a surface whose residue fields are finite. When the order of $\beta$ is a prime >5, we recover most of standard surface theory including existence of terminal resolutions, Castelnuovo contraction and Zariski factorisation. However, interesting new examples of terminal singularities and Castelnuovo contractions appear, which have no characteristic zero analogue.

Benoit Claudon (Université de Rennes 1)
2022-05-17 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/211/
Title: Numerical characterization of complex tori (https://researchseminars.org/talk/ZAG/211/)
by Benoit Claudon (Université de Rennes 1) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
In this talk I will explain how to recognize complex tori among Kahler klt spaces (smooth in codimension 2) in terms of vanishing of Chern numbers. It requires first to define Chern classes on singular spaces (a rather unstable notion). On the way, we will establish a singular version of the Bogomolov--Gieseker inequality for stable sheaves and study what can be said in the equality case. Joint work with Patrick Graf and Henri Guenancia.

Benjamin Bakker (University of Illinois at Chicago)
2022-05-24 22:00 - 23:00 | https://researchseminars.org/talk/ZAG/212/
Title: Period integrals of algebraic varieties (https://researchseminars.org/talk/ZAG/212/)
by Benjamin Bakker (University of Illinois at Chicago) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Period integrals on complex algebraic varieties are the integrals of algebraic differential forms along topological cycles. They are at the heart of Hodge theory. In this talk I will survey some recent results on the behavior of the functions obtained by taking period integrals in algebraic families, including their transcendence, their relation to the derivatives of the associated period map, and some geometric applications. This is joint work with J. Pila and J. Tsimerman.

Anne-Sophie Kaloghiros (Brunel University)
2022-05-19 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/213/
Title: 1-dimensional K-moduli spaces of Fano 3-folds (https://researchseminars.org/talk/ZAG/213/)
by Anne-Sophie Kaloghiros (Brunel University) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
Recent advances in the study of K-stability have shown that there is a projective good moduli space of K-polystable Q-Gorenstein smoothable Q-Fano varieties of dimension n and volume V. The classification of smooth Fano 3-folds is due to Iskovshikh, Mori and Mukai and dates back to the 80s. While the classification is non-modular, it contains rich information on the geometry of Fano 3-folds. Fano 3-folds offer a rich source of examples in which we can explicitly construct and understand some K-moduli spaces. In this talk, I will discuss some examples of 1-dimensional K-moduli spaces of Fano 3-folds. This is joint work with Abban, Cheltsov, Jiao, Martinez-Garcia, Papazachariou and Sarikyan.

Evgeny Shinder (University of Sheffield)
2022-05-26 14:00 - 15:00 | https://researchseminars.org/talk/ZAG/214/
Title: Motivic invariants of birational maps (https://researchseminars.org/talk/ZAG/214/)
by Evgeny Shinder (University of Sheffield) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
I will introduce invariants of birational maps with values in the Grothendieck ring of varieties and their applications to groups of birational isomorphisms, in particular the Cremona groups. This is joint work with Hsueh-Yung Lin.

Nicholas Shepherd-Barron (King’s College London)
2022-05-31 10:00 - 11:00 | https://researchseminars.org/talk/ZAG/215/
Title: Periods of elliptic surfaces (https://researchseminars.org/talk/ZAG/215/)
by Nicholas Shepherd-Barron (King’s College London) as part of ZAG (Zoom Algebraic Geometry) seminar

Abstract
The derivative of the period map for elliptic surfaces has a description in terms of where the j-invariant ramifies. This leads to a natural orthonormal basis of the (1,1) part of the primitive cohomology, expressed in terms of meromorphic 2-forms of the 2nd kind, and to a generic Torelli theorem. This extends to certain other families of varieties fibred in Calabi-Yau varieties.
