Moduli of fibered surface pairs from twisted stable maps

Abstract

Abstract In this paper, we use the theory of twisted stable maps to construct compactifications of the moduli space of pairs

$$(X \rightarrow C, S + F)$$

( X → C , S + F )

where

$$X \rightarrow C$$

X → C

is a fibered surface, S is a sum of sections, F is a sum of marked fibers, and

$$(X,S+F)$$

( 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 with no marked fibers. Furthermore, we compare our compactification to Alexeev’s space of stable maps and the KSBA compactification. As an application, we describe the boundary of a compactification of the moduli space of elliptic surfaces.