A proof of the Göttsche-Yau-Zaslow formula
By Yu-jong Tzeng. 2010
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Let S be a complex smooth projective surface and L be a line bundle on S. G\"ottsche 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 G\"ottsche'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 G\"ottsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasi-modular forms and two unknown series.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Pharmacology
View Paper: https://www.semanticscholar.org/paper/d822522f26d07d3967c4b5db7508a0c6ca00042f
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Pharmacology
Citations in the corpus
- Refined curve counting on complex surfaces (2012, background): “The above formula is by now a theorem, since the existence of universal formulas has been established [70; 43], and the case of the K3 surface (where KS D 0 hence the Bi do not appear) was solved explicitly; see Bryan and Leung [10].”