Moduli space of complete stable pairs

TL;DR

This paper introduces the concept of complete stable pairs on smooth projective varieties, constructs their moduli space, and explores their geometric properties, including morphisms to known moduli spaces and explicit examples on projective lines.

Contribution

The paper defines complete stable pairs and constructs their moduli space, providing new insights into their structure and relationships with existing moduli spaces.

Findings

Moduli space of complete stable pairs constructed

Natural morphisms to stable pairs and Quot-schemes established

On $\,\mathbb{P}^1$, the moduli space is an iterated blow-up of stable pairs

Abstract

We define complete stable pairs on a smooth projective variety, and construct their moduli space. These moduli spaces have natural morphisms to the moduli of stable pairs and Quot-schemes. As an example, we show that the moduli of complete stable pairs on $\mathbb P^1$ is an iterated blowing-up of the moduli of stable pairs, similar to the construction of the space of complete collineations.