Moduli of surfaces fibered in log Calabi-Yau pairs

Abstract

We study the moduli spaces of surface pairs $(X,D)$ admitting a log Calabi--Yau fibration $(X,D)\to C$. We develop a series of results on stable reduction and apply them to give an explicit description of the boundary of the KSBA compactification. Three interesting cases where our results apply are: (1) divisors on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ of bidegree $(2n,m)$; (2) K3 surfaces which map $2:1$ to ${\mathbb{F}}_n$, with $X={\mathbb{F}}_n$ and $D$ the ramification locus, or (3) elliptic surfaces with either a section or a bisection. The main tools employed are stable quasimaps, the canonical bundle formula, and the minimal model program.

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