Steven L. Kleiman
Published 2013-02-06Version 1
Let n_\delta be the number of \delta-nodal curves lying in a suitably ample complete linear system |L| and passing through appropriately many points on a smooth projective complex algebraic surface. A major open problem is to understand the behavior of n_\delta, specifically to finish off Lothar G\"ottsche's mostly proved 1997 conjectures and then go on to treat the new refinements by G\"ottsche and Vivek Shende. Five subproblems are explained, and work on them is surveyed.
Comments: Final version of a 5 page writeup of the author's 20 minute talk at the problem session, 25 Aug 2012, at the conference in honor of Joseph Harris's 60th birthday -- to appear in a volume of the Clay series (put out by the American Mathematical Society)
Curve counting and S-duality
Curve counting on abelian surfaces and threefolds
Curve counting on elliptically fibered Calabi-Yau 3-folds
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