University of Utah
Algebraic Geometry Seminar
Spring 2010: Tuesdays 3:30-4:30, LCB 215

Schedule of upcoming talks:

Click on the title of a talk for the abstract (if available).

Date Speaker Title
January 12
Ilya Shapiro
(MPIM, Bonn)
Chiral Hecke algebra and representations of affine Kac-Moody Lie algebras I will discuss the conjectural role that the Chiral Hecke algebra plays in the local geometric Langlands correspondence. If time permits, I will sketch a possible approach to the unramified case using BRST reduction.
January 19
Yu-jong Tzeng
(Stanford University)
The Universal Formulas for Counting Nodal Curves on Surfaces The problem of counting nodal curves on algebraic surfaces has been studied since the nineteenth century. On the projective surface, it asks how many curves defined by homogeneous degree d polynomials have only nodes as singularities and pass through points in general position. On K3 surfaces, the number of rational nodal curves was predicted by the famous Yau-Zaslow formula. Goettsche conjectured that for sufficiently ample line bundles L on algebraic surfaces, the numbers of nodal curves in |L| are given by universal polynomials in four topological numbers. Furthermore, based on the Yau-Zaslow formula he gave a conjectural generating function in terms of quasi-modular forms. The formula is consistent with many existing results on projective surface, K3, and curves with at most 8 nodes on general surfaces. In this talk, I will discuss how degeneration methods can be applied to count nodal curves and sketch my proof of Goettsche's conjecture.
January 20
LCB 222, 3:30-4:30pm
Chenyang Xu
The automorphic groups of general type varieties (Joint with C. Hacon and J. McKernan) In this talk, I will sketch a work-in-progress, which is aimed to show that the order of the automorphic group of a general type varietie, which is well-known to be finite, is linearly bounded by the volume of K_X. This is a generalization of the classical Hurwitz's theorem in the 1 dimensional case, and G. Xiao's theorem in the 2 dimensional case.
January 26 No seminar today
February 2 Y.P. Lee
Algebraic Cobordism of Bundles on Varieties
February 9 No seminar today
See also: February 12
Comm. Algebra Seminar
Hai Long Dao
(University of Kansas)
Non-commutative desingularizations and cluster-tilting objects over Cohen-Macaulay rings Let M be a reflexive module over a Cohen-Macaulay local ring R and A = Hom(M,M). If A has finite global dimension, then it has been proposed to serve as a non-commutative analogue of desingularizations of Spec(R). In this talk, we will survey what is known about when such module M exists, and discuss the connections with birational geometry and representation theory of commutative rings. Part of the new results are joint work with Craig Huneke.
February 16 No seminar today
February 24
LCB 222, 4:30-5:30pm
Sébastien Boucksom
(CNRS, Paris)
Filtered linear series and Okounkov bodies I will present joint work with Huayi Chen where we associate to certain filtrations on the ring of sections of a big line bundle on a projective variety a concave function on the Okounkov body of the line bundle, in such a way that the image of the Lebesgue measure under this function describes the asymptotic distribution of the jumps of the filtration. Combining this with results of Gillet-Soulé we get a natural and simple proof of the existence of the arithmetic volume as a limit as well as the Fujita approximation theorem in the context of Arakelov geometry.
March 2 Aaron Bertram
Quiver moduli are Bridgeland moduli on the projective plane Le Potier constructed the moduli spaces of stable torsion-free sheaves on P^2 by converting them (via the Beilinson construction) into representations of a quiver. This conversion generalizes to allow geometric invariant theory constructions of all moduli spaces of Bridgeland-stable objects on P^2. This talk is based on work of Bayer and Macri.
March 9
Justin Sawon
(University of North Carolina)
Classifying maps of Lagrangian fibrations Fibrations on holomorphic symplectic manifolds are extremely restricted. Matsushita proved that the fibres must be complex tori which are Lagrangian with respect to the holomorphic symplectic form. When the total space X is projective, Hwang proved that the base of a fibration must be projective space P^n. We therefore obtain a map from (an open subset of) P^n into the moduli space of n-dimensional abelian varieties. In this talk I will describe how this "classifying map" can be used to obtain finiteness results for the number of Lagrangian fibrations, up to deformation.
March 10
LCB 323, 2:00-3:00pm
Arend Bayer
(University of Connecticut)
Stability conditions for the local projective plane I will talk about the space of stability conditions for the total space of the cotangent bundle on P^2. Its geometry is related to classical questions on Chern classes of stable vector bundles P^2, to the modular group Gamma_1(3), and, via mirror symmetry, to the moduli space of elliptic curves with Gamma_1(3)-level structure. This is based on joint work with Emanuele Macri.
March 16 Ching-Jui Lai
Varieties fibered by good minimal models
March 30
Luis Hernández Lamoneda
(CIMAT, Guanajuato, Mexico)
Curvature restricitions for (almost-) complex manifolds Consider the following open questions: -Is S^6 a complex manifold? -If a compact M admits a hyperbolic structure, can it be also a complex manifold? Perhaps the answer to both questions is no. And perhaps the reason it is so is because these manifolds admit metrics of constant non-zero curvature. In studying a (almost-) complex manifold, there is no loss in assuming that it also carries a compatible Riemannian metric (compatibility in the sense that the almost-complex structure is orthogonal with respect to the metric). With it one can use tools of Riemannian geometry to study these objects. In this talk I will give (very) partial answers to the above questions, which ultimately do depend on the choice of particular metrics. Nevertheless, they could provide some insight into the general questions.
April 6
Dustin Cartwright
(University of California, Berkeley)
Mustafin Varieties A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in the Bruhat-Tits tree of PGL(2) and in the two-dimensional building of PGL(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types.
April 13 Rob Easton
Riemann-Roch Theorem for graphs and tropical curves
April 20 Stefano Urbinati
Discrepancies on non-Q-Gorenstein varieties I am going to give an example of a non-Q-Gorenstein variety which is canonical but not klt, and whose canonical divisor has an irrational valuation. I will also give an example of an irrational jumping number and I will prove that there are no accumulation points for the jumping numbers of normal non-Q-Gorenstein varieties with isolated singularities.

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