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

We study the Abramovich–Vistoli moduli space of genus zeroorbifold stable maps to $[\Sym^2 \PP^2]$, the stacksymmetric square of $\PP^2$. This space compactifies themoduli space of stable maps from hyperelliptic curves to$\PP^2$, and we show that all genus zero\break Gromov–Witteninvariants are determined from trivial enumerative geometryof hyperelliptic curves. We also show how the genus zeroGromov–Witten invariants can be used to determine thenumber of hyperelliptic curves of degree $d$ and genus $g$interpolating $3d + 1$ generic points in $\PP^2$.Comparing our method to that of Graber for calculating thesame numbers, we verify an example of the crepantresolution conjecture.

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Published 3 February 2012