Moduli of surfaces fibered in (log) Calabi-Yau pairs II: elliptic surfaces

TL;DR

This paper advances the understanding of the moduli space of elliptic surfaces with a bisection, classifying boundary surfaces and connecting to existing moduli stack results, while avoiding certain MMP techniques.

Contribution

It provides a boundary classification for elliptic surfaces with a bisection and offers a new proof of moduli stack results without using explicit MMP steps.

Findings

Classified boundary surfaces in the KSBA-moduli space of elliptic surfaces with a bisection.

Reproduced known moduli stack results for elliptic surfaces with a section.

Achieved compactification of the moduli stack of hyperelliptic K3 surfaces.

Abstract

This paper continues the study initiated in [ISZ25] on the moduli of surfaces admitting lc-trivial fibrations. Using the techniques developed in [ISZ25], we (1) provide a classification of the surfaces appearing on the boundary of the KSBA-moduli space of elliptic surfaces with a bisection (2) recover the results of a series of papers on the moduli stacks of elliptic surfaces with a section [AB22, Inc20, Bru15]. Notably, our proof of (2) avoids the use of explicit steps of an MMP, such as the "La Nave flip" from [LN02], which plays a central role in [AB22,Inc20]. As an application, we compactify the moduli stack of hyperelliptic K3 surfaces.