Curves with colliding points: logarithmic and stacky

TL;DR

This paper introduces generalized log twisted curves with marked points, explores their moduli and contraction maps, and connects to twisted stable maps, expanding the theoretical framework for algebraic curves.

Contribution

It defines a new class of curves called generalized log twisted curves, extending existing notions and analyzing their moduli and contraction maps.

Findings

Generalized log twisted curves are well-defined and relate to twisted curves when markings are distinct.

The moduli space of these curves is studied and characterized.

Contraction maps between such curves are analyzed for their properties.

Abstract

We introduce a new notion of generalized log twisted curves, which are marked nodal curves with additional data at the marked points. In the case when the markings are distinct this notion agrees with the notion of twisted curve introduced by Abramovich and Vistoli. In addition to developing the basic notions and results, we study in this article the moduli of such curves as well as contraction maps between them. This is motivated, in part, by applications to twisted stable maps which will be studied in a subsequent article.