Moduli of stable maps in genus one and logarithmic geometry I

TL;DR

This paper develops a logarithmic geometric framework for genus one stable maps, constructing smooth moduli spaces and providing insights into the log minimal model program for elliptic curves.

Contribution

It introduces a new approach to moduli of genus one stable maps using logarithmic structures, including smooth proper spaces and interpretations of desingularizations.

Findings

Constructed smooth, proper moduli spaces for genus one stable maps.

Provided a modular interpretation of Vakil and Zinger's desingularization.

Applied methods to the log minimal model program for elliptic curves.

Abstract

This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus $1$. We construct a smooth and proper moduli space dominating the main component of Kontsevich's space of stable genus $1$ maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger's famous desingularization of the Kontsevich space of maps in genus $1$. Our methods also lead to smooth and proper moduli spaces of pointed genus $1$ quasimaps to projective space. Finally, we present an application to the log minimal model program for $\mathcal{M}_{1,n}$. We construct explicit factorizations of the rational maps among Smyth's modular compactifications of pointed elliptic curves.