Open Access
March 2012 A proof of the Göttsche-Yau-Zaslow formula
Yu-Jong Tzeng
J. Differential Geom. 90(3): 439-472 (March 2012). DOI: 10.4310/jdg/1335273391


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.


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Yu-Jong Tzeng. "A proof of the Göttsche-Yau-Zaslow formula." J. Differential Geom. 90 (3) 439 - 472, March 2012.


Published: March 2012
First available in Project Euclid: 24 April 2012

zbMATH: 1253.14054
MathSciNet: MR2916043
Digital Object Identifier: 10.4310/jdg/1335273391

Rights: Copyright © 2012 Lehigh University

Vol.90 • No. 3 • March 2012
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