Contractions of subcurves of families of log curves

TL;DR

This paper introduces a method for contracting certain subcurves of log curves to produce singularities, preserving genus and compatible with families, facilitating maps between moduli spaces of curves.

Contribution

It defines mesa curves with a logarithmic structure and piecewise linear functions to systematically contract subcurves in families of log curves, including elliptic Gorenstein singularities.

Findings

Constructed contractions of subcurves in families of log curves.

Introduced mesa curves with a new logarithmic and tropical structure.

Produced singularities such as elliptic Gorenstein singularities.

Abstract

Let $C$ be a nodal curve, and let $E$ be a union of semistable subcurves of $C$. We consider the problem of contracting the connected components of $E$ to singularities in a way that preserves the genus of $C$ and makes sense in families, so that this contraction may induce maps between moduli spaces of curves. In order to do this, we introduce the notion of mesa curve, a nodal curve $C$ with a logarithmic structure and a piecewise linear function $\overline{\lambda}$ on the tropicalization of $C$. This piecewise linear function determines a subcurve $E$. We then construct a contraction of $E$ inside of $C$ for families of mesa curves. Resulting singularities include the elliptic Gorenstein singularities.