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 $\left| L\right|$ 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 Göttsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasimodular forms and two unknown series.
"A proof of the Göttsche-Yau-Zaslow formula." J. Differential Geom. 90 (3) 439 - 472, March 2012. https://doi.org/10.4310/jdg/1335273391