researchseminars.org

Ana-Maria Castravet (Université Paris-Saclay, UVSQ)
2020-12-07 13:00 - 14:00 UTC | https://researchseminars.org/talk/EDGE2020/1/
Title: Blown-up toric surfaces with non-polyhedral effective cone (https://researchseminars.org/talk/EDGE2020/1/)
by Ana-Maria Castravet (Université Paris-Saclay, UVSQ) as part of EDGE 2020 (online)

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 effective cone, both in characteristic 0 and in prime characteristic. As a consequence, we prove that the 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.

Ruadhaí Dervan (University of Cambridge)
2020-12-07 14:15 - 15:15 | https://researchseminars.org/talk/EDGE2020/2/
Title: The complex geometry of Bridgeland stability conditions (https://researchseminars.org/talk/EDGE2020/2/)
by Ruadhaí Dervan (University of Cambridge) as part of EDGE 2020 (online)

Abstract
I will discuss geometric partial differential equations on holomorphic vector bundles that one can associate to Bridgeland stability conditions. The setup can be seen as a generalisation of the Hitchin-Kobayashi correspondence, which relates slope stability of vector bundles with the existence of Hermite-Einstein metrics. I will then describe some foundational results concerning these new equations. The algebraic geometry involved will be light on technicalities; triangulated categories will not appear. This is joint work with John McCarthy and Lars Sektnan.

Chenyang Xu (Princeton University)
2020-12-07 15:30 - 16:30 | https://researchseminars.org/talk/EDGE2020/3/
Title: On positivity of the CM line bundle (https://researchseminars.org/talk/EDGE2020/3/)
by Chenyang Xu (Princeton University) as part of EDGE 2020 (online)

Abstract
(Joint with Ziquan Zhuang) The K-moduli which parametrizes K-polystable Fano varieties is conjecturally to be projective. In this talk, I will discuss the result we obtained for the positivity of the CM line bundle on the K-moduli. In particular, it implies the (proper) component parametrizing smoothable K-polystable Fano varieties is projective.

Nivedita Viswanathan (University of Edinburgh)
2020-12-08 13:00 - 14:00 | https://researchseminars.org/talk/EDGE2020/4/
Title: On K-stability of some singular del Pezzo surfaces (https://researchseminars.org/talk/EDGE2020/4/)
by Nivedita Viswanathan (University of Edinburgh) as part of EDGE 2020 (online)

Abstract
There has been a lot of development recently in understanding the existence of Kahler-Einstein metrics on Fano manifolds due to the Yau-Tian-Donaldson conjecture, which gives us a way of looking at this problem in terms of the notion of K-stability. In particular, this problem is solved in totality for smooth del Pezzo surfaces by Tian. For del Pezzo surfaces with quotient singularities, there are partial results. In this talk, we will consider singular del Pezzo surfaces which are quasi-smooth, well-formed hypersurfaces in weighted projective space, and understand what we can say about their K-stability. This is joint work with In-Kyun Kim and Joonyeong Won.

Carolina Araujo (IMPA)
2020-12-08 14:15 - 15:15 | https://researchseminars.org/talk/EDGE2020/5/
Title: Birational geometry of Calabi-Yau pairs and 3-dimensional Cremona transformations (https://researchseminars.org/talk/EDGE2020/5/)
by Carolina Araujo (IMPA) as part of EDGE 2020 (online)

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 $K_X+D\sim 0$. Gizatullin's question is about birational properties of the Calabi-Yau pair $(\mathbb{P}^3,D)$. 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 joint work with Alessio Corti and Alex Massarenti.

Constantin Shramov (Higher School of Economics)
2020-12-08 15:30 - 16:30 | https://researchseminars.org/talk/EDGE2020/6/
Title: Birational automorphisms of Severi-Brauer surfaces. (https://researchseminars.org/talk/EDGE2020/6/)
by Constantin Shramov (Higher School of Economics) as part of EDGE 2020 (online)

Abstract
In 2009, I.Dolgachev and V.Iskovskikh classified finite subgroups of the birational automorphism groups of the projective plane over an algebraically closed field of characteristic zero. I will explain an analog of their result for birational automorphism groups of Severi-Brauer surfaces, i.e., surfaces that become isomorphic to the projective plane after passing to the algebraic closure of the base field. The classification is obtained by using geometric techniques based on the Minimal Model Program together with some theory of central simple algebras.

Victoria Hoskins (Radboud University Nijmegen)
2020-12-09 13:00 - 14:00 | https://researchseminars.org/talk/EDGE2020/7/
Title: Constructing new moduli space via non-reductive geometric invariant theory (https://researchseminars.org/talk/EDGE2020/7/)
by Victoria Hoskins (Radboud University Nijmegen) as part of EDGE 2020 (online)

Abstract
There are some moduli problems where non-reductive groups naturally appear: for example, moduli of hypersurfaces in toric varieties, which may have non-reductive automorphism groups, and moduli of certain unstable objects, such as vector bundles of fixed Harder-Narasimhan type, where there is naturally a parabolic group action. In this talk, I will give an introduction to non-reductive GIT and explain how to construct quotients when the unipotent radical is 'graded' by a copy of the multiplicative group. I will then report on joint work in progress with G. Berczi, J. Jackson and F. Kirwan on the construction of moduli spaces of sheaves of fixed Harder-Narasimhan type.

Gergely Bérczi (Aarhus University)
2020-12-09 15:30 - 16:30 | https://researchseminars.org/talk/EDGE2020/8/
Title: Enumerative applications of non-reductive GIT (https://researchseminars.org/talk/EDGE2020/8/)
by Gergely Bérczi (Aarhus University) as part of EDGE 2020 (online)

Abstract
Polynomial reparametrisation groups form the symmetries of jets of holomorphic curves in complex manifolds. They play central role in various classical problems in geometry. I report on recent work in two, seemingly unrelated questions: (i) degeneracy loci of holomorphic maps between complex manifolds and Thom polynomials of singularities and (ii) the Green-Griffiths-Land and Kobayashi hyperbolicity conjectures. I will explain why moduli of jets is a link between the two, and how recently developed intersection theory of non-reductive GIT quotients led to the proof of the polynomial Kobayashi conjecture, and resulted in new formulas for Thom polynomials.

Eloise Hamilton (University of Oxford)
2020-12-09 14:15 - 15:15 | https://researchseminars.org/talk/EDGE2020/9/
Title: Cohomology of GIT quotients, reductive and non-reductive (https://researchseminars.org/talk/EDGE2020/9/)
by Eloise Hamilton (University of Oxford) as part of EDGE 2020 (online)

Abstract
Geometric Invariant Theory (GIT) is a powerful tool not only for constructing quotients in algebraic geometry, but also for studying the geometry of these quotients. The aim of this talk is to explain how to calculate the (rational) cohomology of quotients constructed on the one hand using classical GIT, and on the other using a recent generalisation of GIT, called Non-Reductive GIT. As its name suggests, Non-Reductive GIT enables the construction of quotients by a certain class of non-reductive group actions. After reviewing existing methods for computing the Poincare series of classical GIT quotients when the initial variety is smooth, we will show how similar methods can be used to compute the Poincare series of non-reductive GIT quotients.

Sébastien Boucksom (École Polytechnique)
2020-12-10 13:00 - 14:00 | https://researchseminars.org/talk/EDGE2020/10/
Title: Non-Archimedean pluripotential theory and K-stability (https://researchseminars.org/talk/EDGE2020/10/)
by Sébastien Boucksom (École Polytechnique) as part of EDGE 2020 (online)

Abstract
Non-Archimedean pluripotential theory interprets test configurations as plurisubharmonic functions on a space of valuations, and provides a powerful analytic framework to study and compare various completions of the set of test configurations, phrased in terms of filtrations, valuations, etc... I will review the main features of this theory, which is joint work with Mattias Jonsson.

Cristiano Spotti (Aarhus University)
2020-12-10 14:15 - 15:15 | https://researchseminars.org/talk/EDGE2020/11/
Title: On relations between K-moduli and symplectic geometry (https://researchseminars.org/talk/EDGE2020/11/)
by Cristiano Spotti (Aarhus University) as part of EDGE 2020 (online)

Abstract
A natural intriguing question is the following: how much the moduli spaces of certain polarized varieties know about the symplectic geometry of the underneath manifold? After giving an overview, I will discuss joint work with T. Baier, G. Granja and R. Sena-Dias where we investigate some relations between the topology of the moduli spaces of certain varieties, of the symplectomorphism group and of the space of compatible integrable complex structures. In particular, using results of J. Evans, we show that the space of such complex structures for monotone del Pezzo surfaces of degree four and five is weakly homotopically contractible.

Giulio Codogni (Università degli Studi Tor Vergata)
2020-12-10 15:30 - 16:30 | https://researchseminars.org/talk/EDGE2020/12/
Title: Ample cone of KSB moduli spaces and higher dimensional slope inequalities (https://researchseminars.org/talk/EDGE2020/12/)
by Giulio Codogni (Università degli Studi Tor Vergata) as part of EDGE 2020 (online)

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

Milena Hering (University of Edinburgh)
2020-12-11 13:00 - 14:00 | https://researchseminars.org/talk/EDGE2020/13/
Title: T-Stability of Toric Tangent bundles (https://researchseminars.org/talk/EDGE2020/13/)
by Milena Hering (University of Edinburgh) as part of EDGE 2020 (online)

Abstract
In this talk I will give a brief introduction to slope stability and present a combinatorial criterion for the tangent bundle on a polarised toric variety to be stable in terms of the lattice polytope corresponding to the polarisation. I will then give some applications to toric surfaces and toric varieties of Picard rank 2. This is joint work with Benjamin Nill and Hendrik Süss

Andrea Petracci (Freie Universität Berlin)
2020-12-11 14:15 - 15:15 | https://researchseminars.org/talk/EDGE2020/14/
Title: Toric geometry and singularities on K-moduli (https://researchseminars.org/talk/EDGE2020/14/)
by Andrea Petracci (Freie Universität Berlin) as part of EDGE 2020 (online)

Abstract
An immediate consequence of Kodaira-Akizuki-Nakano vanishing is that smooth Fano varieties have unobstructed deformations. The same holds for singular Fano varieties with mild singularities and small dimension.
In this talk I will show how to use the combinatorics of lattice polytopes to construct examples of K-polystable toric Fano varieties with obstructed deformations, dimension at least 3, and canonical singularities. This method produces singularities (even reducible and non-reduced) on K-moduli stacks and K-moduli spaces of Fano varieties.
This is joint work with Anne-Sophie Kaloghiros.

Hendrik Süß (University of Manchester)
2020-12-11 15:30 - 16:30 | https://researchseminars.org/talk/EDGE2020/15/
Title: Beta-invariants on T-varieties (https://researchseminars.org/talk/EDGE2020/15/)
by Hendrik Süß (University of Manchester) as part of EDGE 2020 (online)

Abstract
In my talk I am presenting previous results on the K-stability of T-varieties in the new packaging of beta-invariants. The main result will be that for T-varieties of complexity 1 Kento Fujita's divisorial polystability is (almost) equivalent to K-polystability. One practical implication will be that for testing K-polystability of such T-varieties it is sufficient to calculate beta-invariants only for a finite number number of "special" T-invariant prime divisors on the variety itself.
