# Elliptic Gromov-Witten Invariants of Del-Pezzo Surfaces

## TL;DR

This paper derives a recursive formula for counting genus one curves of a given degree on del-Pezzo surfaces passing through generic points, using cohomology relations and Gromov-Witten invariants.

## Contribution

It introduces a new recursive approach based on Getzler's relations to compute genus one Gromov-Witten invariants of del-Pezzo surfaces, complementing previous methods.

## Key findings

- Derived a recursive formula for genus one invariants
- Validated the formula with low degree checks
- Compared results with previous methods

## Abstract

We obtain a formula for the number of genus one curves with a variable 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 using Getzler's relationship among cohomology classes of certain codimension 2 cycles in $\overline{M}_{1,4}$ and recursively computing the genus-one Gromov-Witten invariants of del Pezzo surfaces. Using completely different methods, this problem has been solved earlier by Bertram and Abramovich, Ravi Vakil, Dubrovin and Zhang and more recently using Tropical geometric methods by M. Shoval and E. Shustin. We also subject our formula to several low degree checks and compare them to the numbers obtained by the earlier authors.