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

Abstract

In this paper, we study the Gromov–Witten theory of the Hilbert scheme [Formula: see text] of two points on an elliptic surface [Formula: see text]. Assume that [Formula: see text] contains an element supported on the smooth fibers of [Formula: see text]. 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 [Formula: see text]-point genus-[Formula: see text] Gromov–Witten invariant [Formula: see text] up to some rational number [Formula: see text] depending only on [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] is a smooth fiber of [Formula: see text], [Formula: see text] with [Formula: see text] being a fixed point, and [Formula: see text]. Moreover, we propose a conjecture regarding [Formula: see text], and prove that the conjecture is true for [Formula: see text] where [Formula: see text] is an elliptic curve and [Formula: see text] is a smooth curve.