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

TL;DR

This paper derives a formula for counting genus one curves with fixed complex structure on del-Pezzo surfaces, linking symplectic invariants and intersection theory on moduli spaces.

Contribution

It provides a novel enumerative formula for genus one curves with fixed complex structure on del-Pezzo surfaces, connecting symplectic invariants to intersection numbers.

Findings

Derived explicit formula for genus one curve counts

Connected symplectic invariants with intersection theory

Applied to enumerative geometry of del-Pezzo surfaces

Abstract

We obtain a formula for the number of genus one 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 enumerative problem is expressed as the difference between the symplectic invariant and an intersection number on the moduli space of rational curves.