Geometry & Topology
- Geom. Topol.
- Volume 15, Number 1 (2011), 397-406.
A short proof of the Göttsche conjecture
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.
Geom. Topol., Volume 15, Number 1 (2011), 397-406.
Received: 2 November 2010
Accepted: 12 December 2010
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14C05: Parametrization (Chow and Hilbert schemes) 14N10: Enumerative problems (combinatorial problems)
Secondary: 14C20: Divisors, linear systems, invertible sheaves 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Kool, Martijn; Shende, Vivek; Thomas, Richard P. A short proof of the Göttsche conjecture. Geom. Topol. 15 (2011), no. 1, 397--406. doi:10.2140/gt.2011.15.397. https://projecteuclid.org/euclid.gt/1513732280