Genus two enumerative invariants in del-Pezzo surfaces with a fixed complex structure

TL;DR

This paper derives a formula for counting genus two curves with fixed complex structure on del-Pezzo surfaces, extending symplectic methods to relate invariants and intersection numbers.

Contribution

It introduces a novel formula for genus two enumerative invariants on del-Pezzo surfaces using an extended symplectic approach.

Findings

Derived a formula for genus two curves with fixed complex structure

Connected symplectic invariants to intersection theory on moduli space

Extended symplectic methods to higher genus enumerative problems

Abstract

We obtain a formula for the number of genus two curves with a fixed complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This is done by extending the symplectic approach of Aleksey Zinger. This enumerative problem is expressed as the difference between the symplectic invariant and an intersection number on the moduli space of rational curves on the surface.