The genus zero Gromov-Witten invariants of [Sym^2 P^2]

TL;DR

This paper computes genus zero Gromov-Witten invariants of the symmetric square of P^2, linking them to hyperelliptic curve enumeration and verifying aspects of the crepant resolution conjecture.

Contribution

It introduces a method to determine genus zero Gromov-Witten invariants of [Sym^2 P^2] using hyperelliptic curve geometry, connecting enumerative geometry with orbifold Gromov-Witten theory.

Findings

All genus zero Gromov-Witten invariants are derived from hyperelliptic curve geometry.

The invariants determine the count of hyperelliptic curves passing through generic points.

Verification of the crepant resolution conjecture in a specific example.

Abstract

We study the Abramovich--Vistoli moduli space of genus zero orbifold stable maps to [Sym^2 P^2], the stack symmetric square of P^2. This compactifies the moduli space of stable maps from hyperelliptic curves to P^2, 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^2. Comparing our method to that of Graber for calculating the same numbers, we verify an example of the crepant resolution conjecture.