Yu-Jong Tzeng : Counting curves with higher singularities on surfaces

Posted by Mihnea Popa , part of the Algebraic Geometry Seminar.

At

Nov. 9, 2011, 4 p.m.

In

SEO 427

Abstract

A famous problem in classical algebraic geometry is how many r-nodal curves are there in a linear system |L| on an algebraic surface S. If the line bundle L is sufficiently ample, Gottsche conjectured that the number of r-nodal curves is a universal polynomial of Chern numbers of L and S for any r. This conjecture was proven independently by Tzeng and Kool-Shende-Thomas In this talk we will generalize Gottsche's conjecture and show the numbers of curves with any number of arbitrary isolated singularity on surfaces are also given by universal polynomials. Moreover these polynomials can be combined to form a huge generating series and we will discuss its properties.