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SUMMARY:Victor Batyrev (University of TÃ¼bingen)
DTSTART;VALUE=DATE-TIME:20210611T120000Z
DTEND;VALUE=DATE-TIME:20210611T130000Z
DTSTAMP;VALUE=DATE-TIME:20210612T232226Z
UID:ToricDeg/1
DESCRIPTION:Title: Variations on the theme of classical discriminant\nby Victor Batyrev
(University of TÃ¼bingen) as part of Toric Degenerations\n\n\nAbstract\nTh
e classical discriminant $\\Delta_n(f)$ of a degree $n$ polynomial $f(x)$
is an irreducible homogeneous polynomial of degree $2n-2$ on the coefficie
nts $a_0\, \\ldots\, a_n$ of $f$ that vanishes if and only if $f$ has a m
ultiple zero. I will explain a tropical proof of the theorem of Gelfand\,
Kapranov and Zelevinsky (1990) that identifies the Newton polytope $P_n$
of $\\Delta_n$ with a $(n-1)$-dimensional combinatorial cube obtained from
the classical root system of type $A_{n-1}$. Recently Mikhalkin and Tsikh
(2017) discovered a nice factorization property for truncations of $\\Del
ta_n$ with respect to facets $\\Gamma_i$ of $P_n$ containing the vertex $v
_0 \\in P_n$ corresponding to the monomial $a_1^2 \\cdots a_{n-1}^2 \\in
\\Delta_n$. I will give a GKZ-proof of this property and show its connecti
on to the boundary stata in the $(n-1)$-dimensional toric Losev-Manin modu
li space $\\overline{L_n}$. Some variations on the above statements will b
e discussed in connection to the toric moduli space associated with the ro
ot system of type $B_n$ and the mirror symmetry for $3$-dimensional cyclic
quotient singularities ${\\mathbb C}^3/\\mu_{2n+1}$.\n
LOCATION:https://researchseminars.org/talk/ToricDeg/1/
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SUMMARY:Lara Bossinger (UNAM Oaxaca)
DTSTART;VALUE=DATE-TIME:20210611T131500Z
DTEND;VALUE=DATE-TIME:20210611T141500Z
DTSTAMP;VALUE=DATE-TIME:20210612T232226Z
UID:ToricDeg/2
DESCRIPTION:Title: Newton--Okounkov bodies for cluster varieties\nby Lara Bossinger (UNA
M Oaxaca) as part of Toric Degenerations\n\n\nAbstract\nCluster varieties
are schemes glued from algebraic tori. Just as tori themselves\, they come
in dual pairs and it is good to think of them as generalizing tori. Just
as compactifications of tori give rise to interesting varieties\, (partial
) compactifications of cluster varieties include examples such as Grassman
nians\, partial flag varieties or configurations spaces. A few years ago G
ross--Hacking--Keel--Kontsevich developed a mirror symmetry inspired progr
am for cluster varieties. I will explain how their tools can be used to ob
tain valuations and Newton--Okounkov bodies for their (partial) compactifi
cations. The rich structure of cluster varieties however can be exploited
even further in this context which leads us to an intrinsic definition of
a Newton--Okounkov body.\nThe theory of cluster varieties interacts beauti
fully with representation theory and algebraic groups. I will exhibit this
connection by comparing GHKK's technology with known mirror symmetry cons
tructions such as those by Givental\, Baytev--Ciocan-Fontanini--Kim--van S
traten\, Rietsch and Marsh--Rietsch.\n
LOCATION:https://researchseminars.org/talk/ToricDeg/2/
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SUMMARY:Chris Manon (University of Kentucky)
DTSTART;VALUE=DATE-TIME:20210611T143000Z
DTEND;VALUE=DATE-TIME:20210611T153000Z
DTSTAMP;VALUE=DATE-TIME:20210612T232226Z
UID:ToricDeg/3
DESCRIPTION:Title: When is a (projectivized) toric vector bundle a Mori dream space?\nby
Chris Manon (University of Kentucky) as part of Toric Degenerations\n\n\n
Abstract\nLike toric varieties\, toric vector bundles are a rich class of
varieties which admit a combinatorial description. Following the classifi
cation due to Klyachko\, a toric vector bundle is captured by a subspace a
rrangement decorated by toric data. This makes toric vector bundles an ac
cessible test-bed for concepts from algebraic geometry. Along these line
s\, Hering\, Payne\, and Mustata asked if the projectivization of a toric
vector bundle is always a Mori dream space. Suess and Hausen\, and Gonza
les showed that the answer is "yes" for tangent bundles of smooth\, projec
tive toric varieties\, and rank 2 vector bundles\, respectively. Then Her
ing\, Payne\, Gonzales\, and Suess showed the answer in general must be "n
o" by constructing an elegant relationship between toric vector bundles an
d various blow-ups of projective spaces\, in particular the blow-ups of ge
neral arrangements of points studied by Castravet\, Tevelev and Mukai. In
this talk I'll review some of these results\, and then give a new descrip
tion of toric vector bundles by tropical information. This description al
lows us to characterize the Mori dream space property in terms of tropical
and algebraic data\, and produce new families of Mori dream spaces indexe
d by the integral points in a locally closed polyhedral complex. Along t
he way I'll discuss plenty of examples and some questions. This is joint
work with Kiumars Kaveh.\n
LOCATION:https://researchseminars.org/talk/ToricDeg/3/
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