Fine compactified moduli of enriched structures on stable curves

TL;DR

This paper extends the concept of enriched structures on stable curves to arbitrary base schemes, proves the moduli problem is representable, and constructs a compactification with a universal property related to Néron models.

Contribution

It provides a new definition of enriched structures over arbitrary bases, proves the representability of the moduli problem, and constructs a compactified stack with a universal property.

Findings

The moduli problem for enriched structures is representable.

A compactification of the stack of enriched structures is constructed.

The resulting object has a universal property related to Néron models.

Abstract

Enriched structures on stable curves over fields were defined by Maino in the late 1990s, and have played an important role in the study of limit linear series and degenerating jacobians. In this paper we solve three main problems: we give a definition of enriched structures on stable curves over arbitrary base schemes, and show that the resulting fine moduli problem is representable; we show that the resulting object has a universal property in terms of N\'eron models; and we construct a compactification of our stack of enriched structures.