# Curve counting in genus one: elliptic singularities & relative geometry

## TL;DR

This paper develops a genus one Gromov--Witten theory for very ample pairs, introducing new invariants related to elliptic singularities and relating relative and absolute theories through degeneration techniques.

## Contribution

It constructs reduced, relative genus one Gromov--Witten invariants for very ample pairs and relates them to absolute invariants via degeneration, generalizing genus zero formulas.

## Key findings

- Introduces a desingularisation of the moduli space of genus one stable maps.
- Establishes formulas relating relative and absolute Gromov--Witten invariants.
- Develops techniques for integrals on logarithmic blowups of moduli spaces.

## Abstract

We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute theories by degeneration of the tangency conditions, and the resulting formulas generalise a well-known recursive calculation scheme put forward by Gathmann in genus zero. The geometric input is a desingularisation of the principal component of the moduli space of genus one logarithmic stable maps to a very ample pair, using the geometry of elliptic singularities. Our study passes through general techniques for calculating integrals on logarithmic blowups of moduli spaces of stable maps, which may be of independent interest.