UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

February 23, 2010

939 Evans Hall

3:45PM: Universal Formulas for Counting Nodal Curves on Surfaces

Yu-jong Tzeng

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 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.

5:00PM: Gorenstein semigroup rings and moduli of points on the line.

Chris Manon

We will talk about the projective coordinate rings of a class of toric varieties which appear as toric degenerations of moduli of points on the projective line. After discussing the combinatorics and geometry unique to the associated polytopes, we will show which of these rings are Gorenstein.

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