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A
short proof of the Göttsche conjecture
Martijn Kool, Vivek Shende and Richard P Thomas |
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Geometry & Topology 15 (2011) 397–406
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Abstract |
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We prove that for a sufficiently ample line bundle on a surface , the number of –nodal curves in a general –dimensional linear system is given by a universal polynomial of degree in the four numbers and . The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of Pandharipande and the third author [J. Amer. Math. Soc. 23 (2010) 267–297] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, Göttsche and Lehn [J. Algebraic Geom. 10 (2001) 81–100]. We are also able to weaken the ampleness required, from Göttsche’s –very ample to –very ample. |
Keywords
Göttsche conjecture, Goettsche conjecture, counting nodal
curves on surfaces
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Mathematical Subject Classification 2000
Primary: 14C05, 14N10
Secondary: 14C20, 14N35
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References |
Publication
Received: 2 November 2010
Accepted: 12 December 2010
Published: 1 March 2011
Proposed: Lothar Göttsche
Seconded: Jim Bryan, Gang Tian
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Authors |
| Martijn Kool | |
| Department of Mathematics Imperial College 180 Queens Gate London SW7 2AZ UK |
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| Vivek Shende | |
| Department of Mathematics Princeton University Princeton NJ 08540 USA |
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| Richard P Thomas | |
| Department of Mathematics Imperial College 180 Queens Gate London SW7 2AZ UK |
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