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

In this paper, we study the Gromov–Witten theory of the Hilbert scheme X [ 2 ] of two points on an elliptic surface X. Assume that | K X | contains an element supported on the smooth fibers of 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 -point genus- 0 Gromov–Witten invariant 〈 w 〉 0 , d (β f − 2 β 2) X [ 2 ] up to some rational number m (d , X) depending only on d and X , where w ∈ H 4 (X [ 2 ] , ℂ) , d ≥ 1 , f is a smooth fiber of X , β f = x 0 + f ⊂ X [ 2 ] with x 0 ∈ X − f being a fixed point, and β 2 = { ξ ∈ 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 × E where E is an elliptic curve and C is a smooth curve. [ABSTRACT FROM AUTHOR]