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Gromov–Witten invariants of Hilbert schemes of two points on elliptic surfaces

Mazen M. Alhwaimel

Department of Mathematics, College of Science, Qassim University, P. O. Box 6644, Buraydah 51452, Saudi Arabia

E-mail Address: alhwaimelm@gmail.com

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and

Zhenbo Qin

https://orcid.org/0000-0002-3753-6509

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

E-mail Address: qinz@missouri.edu

Corresponding author.

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https://doi.org/10.1142/S0129167X22500793Cited by:0 (Source: Crossref)

Abstract

In this paper, we study the Gromov–Witten theory of the Hilbert scheme $X^{[2]}$ $X^{[2]}$ of two points on an elliptic surface $X$ $X$ . Assume that $| K_{X} |$ $| K_{X} |$ contains an element supported on the smooth fibers of $X$ $X$ . By analyzing the degeneracy locus and localized virtual cycle arising from the cosection localization theory of Kiem and Li [Y. Kiem and J. Li, Gromov–Witten invariants of varieties with holomorphic 2-forms, preprint; Y. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025–1050], we determine the $1$ $1$ -point genus- $0$ $0$ Gromov–Witten invariant ${⟨ w ⟩}_{0, d (β_{f} - 2 β_{2})}^{X [2]}$ ${〈 w 〉}_{0, d (β_{f} - 2 β_{2})}^{X [2]}$ up to some rational number $m (d, X)$ $m (d, X)$ depending only on $d$ $d$ and $X$ $X$ , where $w \in H^{4} (X^{[2]}, ℂ)$ , $d \geq 1$ , $f$ is a smooth fiber of $X$ , $β_{f} = x_{0} + f \subset X^{[2]}$ with $x_{0} \in X - f$ being a fixed point, and $β_{2} = {ξ \in X^{[2]} | Supp (ξ) = {x_{0}}}$ . Moreover, we propose a conjecture regarding $m (d, X)$ , and prove that the conjecture is true for $X = C \times E$ where $E$ is an elliptic curve and $C$ is a smooth curve.

Communicated by Oscar Garcia-Prada

Keywords:

AMSC: 14C05, 14N35

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