The genus zero Gromov–Witten invariants of the symmetric square of the plane

Abstract

We study the Abramovich–Vistoli moduli space of genus zero orbifold stable maps to [Sym P], the stack symmetric square of P. This space compactifies the moduli space of stable maps from hyperelliptic curves to P, and we show that all genus zero Gromov–Witten invariants are determined from trivial enumerative geometry of hyperelliptic curves. We also show how the genus zero Gromov–Witten invariants can be used to determine the number of hyperelliptic curves of degree d and genus g interpolating 3d+ 1 generic points in P. Comparing our method to that of Graber for calculating the same numbers, we verify an example of the crepant resolution conjecture.