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

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

    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 wX[2]0,d(βf2β2)wX[2]0,d(βf2β2) up to some rational number m(d,X)m(d,X) depending only on dd and XX, where wH4(X[2],), d1, f is a smooth fiber of X, βf=x0+fX[2] with x0Xf being a fixed point, and β2={ξX[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

    AMSC: 14C05, 14N35