2010/09/27 by Tzeng, Yu-jong · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1009.5371
Let S be a complex smooth projective surface and L be a line bundle on S. Göttsche conjectured that for every integer r, the number of r-nodal curves in |L| is a universal polynomial of four topological numbers when L is sufficiently ample. We prove Göttsche's conjecture using the algebraic cobordism group of line bundles on surfaces and degeneration of Hilbert schemes of points. In addition, we prove the the Göttsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasi-modular forms and two unknown series.