Moduli of elliptic curves in products of projective spaces

TL;DR

This paper constructs a smooth compactification of the moduli space of elliptic curves in products of projective spaces, extending previous desingularization methods and enabling the definition of virtual fundamental classes for genus 1 maps.

Contribution

It introduces a new Vakil--Zinger type desingularization for elliptic curves in products of projective spaces using logarithmic geometry.

Findings

Constructed a smooth compactification of the moduli space.

Extended desingularization techniques to products of projective spaces.

Defined virtual fundamental classes for genus 1 maps.

Abstract

We exhibit a smooth compactification of the moduli space of elliptic curves in a product of projective spaces with tangency along a subset of its toric boundary divisors. This is a Vakil--Zinger type of desingularization for maps to a product of projective spaces using ideas of elliptic singularities and logarithmic geometry, extending the recent work by Ranganathan--Santos-Parker--Wise. We use this to construct the virtual fundamental classes of the spaces of genus 1 maps to a special class of simple normal crossings pairs.