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 X[2]X[2] of two points on an elliptic surface XX. Assume that |KX||KX| contains an element supported on the smooth fibers of XX. 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 11-point genus-00 Gromov–Witten invariant 〈w〉X[2]0,d(βf−2β2)⟨w⟩X[2]0,d(βf−2β2) up to some rational number m(d,X)m(d,X) depending only on dd and XX, where w∈H4(X[2],ℂ), d≥1, f is a smooth fiber of X, βf=x0+f⊂X[2] with x0∈X−f being a fixed point, and β2={ξ∈X[2]|Supp(ξ)={x0}}. Moreover, we propose a conjecture regarding m(d,X), and prove that the conjecture is true for X=C×E where E is an elliptic curve and C is a smooth curve.
Communicated by Oscar Garcia-Prada