Counting curves of any genus on P^2_7

TL;DR

This paper computes Gromov-Witten invariants of any genus for del Pezzo surfaces of degree 2 or higher, using tropical geometry and enumerative techniques to extend known results to all genera.

Contribution

It introduces a new method combining tropical geometry and enumerative formulas to compute invariants for del Pezzo surfaces of degree 2 and 3, extending previous genus-zero results.

Findings

Computed Gromov-Witten invariants for all genera on certain del Pezzo surfaces.

Developed a Caporaso-Harris type formula for counting curves with tangency conditions.

Expressed invariants of the plane blown up at 7 points via invariants of a weak Fano surface.

Abstract

We compute Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge2$. The genus zero invariants have been computed a long ago, Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge3$ have been found by Vakil. We solve the problem in two steps: (1) we consider the plane blown up at 6 points on a conic and one more point outside this conic (weak Fano surface), and, using techniques of tropical geometry, obtain a Caporaso-Harris type formula counting curves of any divisor class and genus subject to arbitrary tangency conditions with respect to the blown up conic, (2) then we express the Gromov-Witten invariants of the plane blown up at 7 points via enumerative invariants of the weak Fano surface, using Vakil's version of Abramovich-Bertram formula.