| Sat 9:00 |
Jessica Sidman |
Syzygies of varieties lying on toric varieties
Suppose that X is a variety in projective space with defining ideal
$I$ contained in $S = k[x_0,..., x_n]. $ Many important geometric and algebraic
features of X can be determined from a minimal free resolution of $S/I$. I will
discuss situations in which X lies on a toric variety and the syzygies of the
toric variety can be used to construct a minimal free resolution of $S/I.$ If
time permits I will also discuss generalizations involving secant varieties. |
| Sat 9:30 |
Evgeny Materov |
Tate resolutions for products of projective spaces
In this talk I will discuss recent work in progress with
David Cox on explicit formulas for Tate resolutions. It is known that
for the $d$-fold Veronese embeddings of projective space $\mathbb P^n$
the diagonal maps of the Tate resolution are given by the Bezoutian of
$n+1$ homogeneous forms of degree $d$ in $n+1$ variables. We are
trying to find similar formulas for the sheaves arivisng from the
generalized Segre-Veronese embeddings of product of projective spaces.
For example, Tate resolutions corresponding to the classical Segre
embedding of $\mathbb P^a \times \mathbb P^b$ of degree $d=(1,1)$ are
related to the notion of hyperdeterminant introduced by Gelfand,
Kapranov and Zelevinsky. Also, some important properties of sheaves
like regularity, duality can be determined from the corresponding Tate
resolution. |
| Sat 10:00 |
Sonja Petrovic |
Algebraic properties of cut ideals
A cut ideal of a graph records the relations among the cuts of the
graph. These toric ideals have been introduced by Sturmfels and
Sullivant who also posed the problem of relating their properties to
the combinatorial structure of the graph. We study a nice collection
of graphs that can be obtained from trees andcycles: the outerplanar
graphs. A correspondence between cut ideals of cycles and a recently
studied class of phylogenetic ideals reveals a quadratic Groebner
basis. I nadditio nthe associated semigroup algebras are
Cohen-Macaulay. Using Stumfels and Sullivant's clique-sum
construction, we show that the cut ideals of all outerplanar graphs
have a squarefree quadratic Groebner basis. Thus the coordinate rings
are Cohen-Macaulay, but not Gorenstein in general.
|
| Sat 10:30 |
Paul Horja |
Triangulated categories and toric varieties
Toric varieties have traditionally provided good examples
for testing proposals in algebraic geometry. In this talk, some
results about the structure of their categories of coherent sheaves
will be presented. Some implications of these ideas for homological
mirror symmetry will also be discussed.
|
| Sat 3:00 |
Ezra Miller |
Multiplier ideals of sums via cellular resolutions
Howald [TAMS, 2001] proved a formula for multiplier ideals of monomial
ideals using resolutions of singularities for toric varieties. Takagi
[Amer. J. Math, 2006], using characteristic p methods, proved a
formula for sums two general ideal sheaves in terms of multiplier
ideals of products. In joint work with Shin-Yao Jow, we unify and
generalize these results by constructing a complex of sheaves that
resolves the multiplier ideal of any finite sum of ideal sheaves in
terms of (direct sums of) multiplier ideals of products. The
resolution we construct is cellular, in the sense that its boundary
maps are encoded by the algebraic chain complex of a certain
polyhedral complex. |
| Sat 3:30 |
Howard Thompson |
On the multiplier ideals of a monomial curve
In this talk we will explore two questions about the
multiplier ideals of monomial curves in affine space: To what extent
can toric geometry be used to compute them? And, if the coefficient
is less than the codimension, are they monomial?
|
| Sat 4:00 |
Christine von Renessee |
Combinatorial descriptions of the Mori cone of a toric variety
The Mori cone of a toric variety can be computed by using the
codimension-one cones (walls) of the corresponding fan. Reid's
results show that i nthe complete simplicial case you can instead use
so called primitive collections to find generators of the Mori cone.
I will review this and then explain how to generalize it to any toric
variety with convex support. we will then look at examples of
non-simplicial fans and some of their simplicial refinements to see if
and how we can use the generators of the Mori cone of the refineements
for the original fans.
|
| Sat 4:30 |
Ivan Soprunov |
Lower bounds for the minimum distance of a toric code
A toric code is a linear evaluation code defined by a
lattice convex polytope P in $\mathbb R^n$. The problem of computing
the minimum distance of a toric code is related to finding the largest
number of zeroes in $(\mathbb F^*_Q)^n$ of sections of globally
generated line bundles on the toric variety associated to $P$. John
Little and Hal Schenck have recently shown that for $n=2$ and large
enough $q$ reducible section have more zeroes in $(\mathbb F_q^*)^2$
than irredicble ones. This inspired us to look at the toric codes
defined by zonotopes. We obtain an upper bound for the number of
zeroes in $(\mathbb $(F_q^*)^n$ of sections associated to zonotopes.
we also prove a general lower bound for the minimum distance of a
toric code in terms of the lattice diameters of the polytope $P$ in
$n$ linearly independent directions.
|
| Sat 5:00 |
Joseph Gubeladze |
Big Witt vectors and K-theory of toric varieties
The Bloch-Stienstra-Weibel action of big Witt vectors on higher
$K$-groups of affine monoid rings provides a deep insight into these
mysterious groups. It was used in the proof of the s. c. nilpotence
conjecture for toric varieties over number fields and, more recently,
played a crucial role in extending the result to all regular
coefficient rings containing $\mathbb Q$. In the talk the main result
will be stated and the globalizatio nproecess for the coeffiicent ring
will be outlined. If time permits, a more refined conjectural
description of $K$-theory of toric varieties, based on the big Witt
vectors, will be discussed.
|
| Sun 8:00 |
Alvaro Pelayo |
Maximal toric packings of symplectic-toric manifolds
We explain how the set of symplectic-toric ball packings of a
symplectic-toric manifold of dimensional at least four admits the
structure of a convex polytope. Using this we will show that for each
$n \geq 2$ and each $\delta \in (0,1)$ there are uncountably many
inequivalent $2n$-dimensional symplectic-toric manifolds with a
maximal toric packing of density $\delta$. This result follows from a
general analysis of how the densities of maximal packings chance while
varying a given symplectic-toric manifold through a family of
symplectic-toric manifolds that are equivariantly diffeomorphic but
not equivariantly symplectomorphic. Our theorem is in contrast with a
previous result of the presenter: up to equivalence, only $(\mathbb C
\mathbb P^1)^2 and $\mathbb C \mathbb P^2)$ admit density one packings
when $n=2$ and only $\mathbb C \mathbb P^n$ admits density one
packings when $n>2$.
|
| Sun 8:30 |
Benjamin Nill |
Combinatorial aspects of mirror symmetry
In this short talk I am going to present results and open
questiosn about the lattice polytopes occuring in the toric mirror
symmetry constructions due to Batyrev and Borisov.
|
| Sun 9:00 |
Val Hower |
Z_2 Hodge spaces of fans
I will introduce the notion of a cosheaf on a fan $\Sigma$
and define the $\mathbb Z_2$ Hodge spaces of $\Sigma$, denoted
$H_{pq}(\Sigma)$. I will then compute $H_{pq}(\Sigma)$ when $\Sigma$
is the normal fan of a reflexive polytope. Finally, I will show how
one can use the $\mathbb Z_2$ Hodge spaces of $\Sigma$ to gain
information about the topology of $X_{\Sigma}(\mathbb R)$ and
$X_{\Sigma}(\mathbb C)$, the real and complex toric varieties
associated to tRhe fan $\Sigma$.
|
| Sun 9:30 |
Eric Katz |
Localization on toric varieties and a new proof of Bernstein's theoremn
Bernstein's theorem is an extension of Bezout's theorem that gives a
bound on the number of intersection points of $n$ hypersurfaces in an
$n$-dimensional algebraic torus. We outline a new proof of
Bernstein's theorem. The proof involves a combinatorial description
of the map from equivariant to ordinary Chow chomology of a toric
variety together with Brion's formula for lattice point enumeration.
The ideas behind this proof are motivated by tropical geometry. This
is joint work with Sam Payne.
|
| Sun 10:00 |
Tom Braden |
A ring structure on intersection cohomology of hypertoric varieties
Hypertoric varieties are complex symplectic varieties which
can be viewed as "quarterionic" analogs of toric varieties. They are
described by the combinatorics of hyperplane arrangements, similarly
to the way toric varieties are described by their moment polyhedra.
We show that an inductive procedure which computes the T-equivariant
intersection cohomology of toric varieties also works essentially
unchanged for hypertoric varieties. Unlike the situation for toric
varieties, however, we are able to define a canonical ring structure
on the intersection cohomology of hypertoric varieties.
|
| Sun 10:30 |
Laura Matusevich |
Weyl closure of hypergeometric systems
I will show that GKZ hypergeometric systems, which are built from
toric ideals, and Horn systems, which are built from lattice basis
ideals, are Weyl closed. This means that such systems are the
differential annihilators of their solution spaces
|
| Sun 3:00 |
Greg Smith |
Quivers and toric varieties
In this talk, we introduce toric ideals in the path algebra of a
quiver and examine the induced GIT constructions for toric varieties. We
will also discuss connections with "resolutions of the diagonal".
|
| Sun 3:30 |
Ben Howard |
A nice projective embedding for the geometric invariant theory quotients (P^1)^n/SL_2
Given an $n$-tuple $w=(w_1,\dots,w_n)$ of positive integers,
we study the moduli space $M_w$ of weighted $n$-tuples of points on
the projective line, modulo automorphisms of the line $L_w=
O(w_1,\dots, w_n) = O(w_1) \boxtimes \dots \boxtimes O(w_n)$
over $(\mathbb P^1)^n$. The projective variety $M_w$ has an explicit
embedding into projective space.
We find that if each $w_i$ is an even integer, the projective
coordinate ring $R_w$ of $M_w$ is particularly nice. The ideal of
$R_w$ admits a quadratic Gr\"obner basis. Further, if each w_i=2$
then R_w$ is Gorenstein, and $M_w$ is a Fano variety. All these
results are obtained by degenerating $R_w$ into a toric algebra
$R'_w$. The ideal of $R'_w$ also has a quadratic Gr\"obner basis, and
$R'_w$ is Gorenstein when each $w_i=2$.
|
| Sun 4:00 |
Angela Gibney |
The moduli space of stable fat pointed curves of genus zero
I will introduce a toric variety, the moduli space
$\overline{M}_{0,\{n_1,\ldots,n_m\}}$, whose points correspond to
stable pointed rational curves where the markings have embedded scheme
structure. When $\sum_{i}n_i=n$, this moduli space of
fat pointed curves arises naturally as a Gr\"obner degeneration of
$\overline{M}_{0,n}$, the moduli space of stable $n$-pointed rational
curves. This is joint work with Diane Maclagan.
|
