The Michigan Mathematical Journal

Gromov–Witten Invariants of the Hilbert Scheme of Two Points on a Hirzebruch Surface

Yong Fu

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In Gromov–Witten theory the virtual localization method is used only when the invariant curves are isolated under a torus action. In this paper, we explore a strategy to apply the localization formula to compute the Gromov–Witten invariants by carefully choosing the related cycles to circumvent the continuous families of invariant curves when there are any. For the example of the two-pointed Hilbert scheme of Hirzebruch surface F1, we manage to compute some Gromov–Witten invariants, and then by combining with the associativity law of (small) quantum cohomology ring, we succeed in computing all 1- and 2-pointed Gromov–Witten invariants of genus 0 of the Hilbert scheme with the help of [13].

Article information

Michigan Math. J., Volume 67, Issue 4 (2018), 675-713.

Received: 5 July 2013
Revised: 4 May 2018
First available in Project Euclid: 3 October 2018

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Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]


Fu, Yong. Gromov–Witten Invariants of the Hilbert Scheme of Two Points on a Hirzebruch Surface. Michigan Math. J. 67 (2018), no. 4, 675--713. doi:10.1307/mmj/1538532103.

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