WAGS Fall 2012

WAGS / Fall 2012

Astracts

University of Utah
October 20-21, 2012


Speaker: Matt Baker Title: Linear series on metrized complexes of algebraic curves Abstract: A metrized complex of algebraic curves over a field K is, roughly speaking, a finite edge-weighted graph G together with a collection of marked complete nonsingular algebraic curves C_v over K, one for each vertex; the marked points on C_v correspond to edges of G incident to v. We will present a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical and tropical Riemann-Roch theorems, together with a semicontinuity theorem for the behavior of the rank function under specialization of divisors from smooth curves to metrized complexes. The statement and proof of the latter result make use of Berkovich's theory of non-archimedean analytic spaces. As an application of the above considerations, we formulate a partial generalization of the Eisenbud-Harris theory of limit linear series to semistable curves which are not necessarily of compact type. This is joint work with Omid Amini.

Speaker: Matt Baker
Title: Linear series on metrized complexes of algebraic curves
Abstract: A metrized complex of algebraic curves over a field K is, roughly speaking, a finite edge-weighted graph G together with a collection of marked complete nonsingular algebraic curves C_v over K, one for each vertex; the marked points on C_v correspond to edges of G incident to v. We will present a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical and tropical Riemann-Roch theorems, together with a semicontinuity theorem for the behavior of the rank function under specialization of divisors from smooth curves to metrized complexes. The statement and proof of the latter result make use of Berkovich's theory of non-archimedean analytic spaces. As an application of the above considerations, we formulate a partial generalization of the Eisenbud-Harris theory of limit linear series to semistable curves which are not necessarily of compact type. This is joint work with Omid Amini.


Speaker: Aise Johan de Jong Title: The Stacks Project Abstract: The stacks project is a long term open source, collaborative project documenting and developing theory on algebraic stacks.


Speaker: Sean Keel Title: Theta functions for K3 surfaces Abstract: I will explain joint work with Gross, Hacking and Siebert where we construct a distinguished toroidal compactification of F_g = moduli of polarized K3 surfaces of degree 2g-2, together with a canonical formal family of polarized surfaces along the full boundary, endowed with canonical theta functions -- a canonical basis of sections of the line bundle, together with a formula for the multiplication in the homogeneous coordinate ring determined by counts of rational curves on the Dolgachev-Voisin mirror family. We expect that this family glues to the universal family, and thus gives an algebraic universal family over the compactification, and a proof of Tyurin's conjecture that the classical theory of theta functions for Abelian surfaces extends to K3 surfaces, at least near the large complex structure limit.

Speaker: Sean Keel
Title: Theta functions for K3 surfaces
Abstract: I will explain joint work with Gross, Hacking and Siebert where we construct a distinguished toroidal compactification of F_g = moduli of polarized K3 surfaces of degree 2g-2, together with a canonical formal family of polarized surfaces along the full boundary, endowed with canonical theta functions -- a canonical basis of sections of the line bundle, together with a formula for the multiplication in the homogeneous coordinate ring determined by counts of rational curves on the Dolgachev-Voisin mirror family. We expect that this family glues to the universal family, and thus gives an algebraic universal family over the compactification, and a proof of Tyurin's conjecture that the classical theory of theta functions for Abelian surfaces extends to K3 surfaces, at least near the large complex structure limit.


Speaker: János Kollár Title: Families of Cartier divisors Abstract: One of the difficulties in the theory of moduli of varieties is the existence of divisors on flat families that are Cartier on every fiber but not Cartier on the total space. We study such examples and prove that this is a problem mostly for relative dimension 2 only. A list of questions can be foud here.


Speaker: Burt Totaro Title: The integral Hodge conjecture for 3-folds Abstract: The Hodge conjecture predicts which rational homology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. In other words, it is about the difference between topology and algebraic geometry. The integral Hodge conjecture, the analogous conjecture for integral homology classes, is false in general. We discuss negative results and some new positive results on the integral Hodge conjecture for 3-folds.

Speaker: Burt Totaro
Title: The integral Hodge conjecture for 3-folds
Abstract: The Hodge conjecture predicts which rational homology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. In other words, it is about the difference between topology and algebraic geometry. The integral Hodge conjecture, the analogous conjecture for integral homology classes, is false in general. We discuss negative results and some new positive results on the integral Hodge conjecture for 3-folds.

Department of Mathematics
University of Utah

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