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We use Pixton's relations to prove a reconstruction theorem for genus 2Gromov-Witten invariants in the style of Kontsevich-Manin (genus 0) and Getzler(genus 1). We also calculate genus 2 (descendant) Gromov-Witten invariants of$\mathbb{P}^2$ blown up at a finite number of points in general position.

Reconstruction theorems for genus 2 Gromov-Witten invariants