Moduli of fibered surface pairs from twisted stable maps

TL;DR

This paper constructs a new compactification of the moduli space of fibered surface pairs with marked fibers using twisted stable maps, extending previous work and analyzing boundary structures of elliptic surfaces.

Contribution

It introduces a novel compactification of fibered surface pairs with marked fibers via twisted stable maps, generalizing prior moduli space constructions.

Findings

Constructed a compactification of the moduli space of fibered surface pairs with marked fibers.

Compared the new compactification to Alexeev's stable maps and KSBA compactification.

Described the boundary of the moduli space for elliptic surfaces.

Abstract

We use the theory of twisted stable maps to Deligne-Mumford stacks to construct compactifications of the moduli space of pairs $(X \to C, S + F)$ where $X \to C$ is a fibered surface, $S$ is a sum of sections, $F$ is a sum of marked fibers, and $(X, S + F)$ is a stable pair in the sense of the minimal model program. This generalizes the work of Abramovich-Vistoli, who compactified the moduli space of fibered surfaces without marked fibers. Furthermore, we compare our compactification to Alexeev's space of stable maps and the KSBA compactification of stable pairs. As an application, we describe the boundary of the compactification of the moduli space of elliptic surfaces.