The genus zero Gromov-Witten invariants of [Sym(2) P(2)] and the enumerative geometry of hyperelliptic curves in P(2)

We study the moduli space of orbifold stable maps to the stack symmetric square of the projective plane, [Sym2 P 2]. Viewing this moduli space as a compactification of the moduli space of hyper-elliptic curves to P2, we express several of the genus zero Gromov-Witten invariants of [Sym2 P2] in terms of the enumerative geometry of hyperelliptic curves in P2. Trivial enumerative geometry of hyperelliptic curves is enough to determine all of the genus zero Gromov-Witten invariants, and from these it is possible to extract information about less trivial enumerative problems about hyperelliptic curves in P 2. For example, we obtain a new algorithm for counting the hyperelliptic curves in P2 of genus g and degree d passing through 3d + 1 points in generic position. Comparing this method with that of Graber, using Hilb2 P 2 to compute the same numbers, we can also verify the relationship predicted by crepant resolution conjecture between the genus zero Gromov-Witten theory of [Sym2 P2] and the genus zero Gromov-Witten theory of Hilb2 P2.