Marked nodal curves with vector fields

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.

53 pages, 11 figures. To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci