Marked nodal curves with vector fields

TL;DR

This paper extends stabilization techniques for nodal curves to those with vector fields, establishing their behavior in families and applying this to degenerations of configuration spaces.

Contribution

It introduces operations on nodal curves with vector fields that are compatible with families and constructs inverse operations, extending classical stabilization results.

Findings

Operations commute with base change in families

Inverse operations are constructed under certain conditions

Degeneration of configuration spaces is shown to be isotrivial

Abstract

We discuss two operations on nodal curves with (logarithmic) vector fields, which resemble the `stabilization' construction in Knudsen's proof that $\bar{\mathcal M}_{g,n+1}$ is the universal curve over $\bar{\mathcal M}_{g,n}$. We prove that both operations work in families (commute with base change). We construct inverse operations under suitable assumptions, which allow us to prove a technical result quite similar to Knudsen's, in the case of curves with vector fields. As an application, we prove that the Losev--Manin compactification of the space of configurations of $n$ points on ${\mathbb P}^1 \backslash \{0,\infty\}$ modulo scaling degenerates isotrivially to a compactification of the space of configurations of $n$ points on ${\mathbb A}^1$ modulo translation, and the natural group actions fit together globally.