# Moduli of Curves of Genus One with Twisted Fields

## TL;DR

This paper introduces a new stack for genus one stable curves with twisted fields, providing a blowup-free desingularization of the moduli space of genus one stable maps, advancing the theory of stacks with twisted fields.

## Contribution

It constructs a smooth Artin stack for genus one curves with twisted fields and proves its isomorphism to a blowup stack, enabling a blowup-free resolution of the moduli space.

## Key findings

- Constructed a smooth Artin stack for genus one curves with twisted fields.
- Proved the stack is isomorphic to a blowup stack of existing moduli.
- Provided a blowup-free desingularization of the moduli space of genus one stable maps.

## Abstract

We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two.