Back to Algebraic Geometry Seminar: Spring 2010
Algebraic Geometry Seminar
Date: Tuesday, Jan 19, 2010
Time: 3:30PM - 4:30PM
Location: LCB 215
Yu-jong Tzeng
Stanford University
Title |
The Universal Formulas for Counting Nodal Curves on Surfaces |
Abstract |
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. |
Algebraic Geometry