The moduli stack of enriched structures and a logarithmic compactification

TL;DR

This paper introduces a logarithmic perspective on enriched curves, defining a moduli stack of rich log curves, and constructs a smooth modular compactification, extending the concept to r-rich curves with connections to tropical geometry.

Contribution

It provides a new logarithmic framework for enriched curves, leading to a simple modular compactification and generalizations to r-rich curves, answering open questions from prior work.

Findings

Defined the stack of rich log curves as an open substack of log curves

Constructed a smooth log blowup compactification of the stack of log curves

Extended the concept to r-rich curves with similar properties

Abstract

Enriched curves have been studied over algebraically closed fields by Main\`o ([Mai98]) and recently over general base schemes in [BH19]. In this paper, we study enriched curves from a logarithmic viewpoint: we give a succinct definition of the stack of rich log curves, which is an open substack of the stack of log curves, and define an enriched curve to be a curve with a minimal rich log structure on it. This logarithmic view point turns out to be a natural language for enriched structures, leading naturally to a simple modular compactification. This modular compactification is a smooth log blowup of the stack of log curves, answering affirmatively two questions from [BH19]. We also generalise the concept of rich curves to $r$-rich curves, and show similar results. We include a chapter phrasing some of the key definitions solely in the language of real tropical geometry.