Moduli of Persson surfaces: The compactification via KSBA stable pairs and a generic global Torelli type theorem

TL;DR

This paper constructs and describes the compactified moduli space of Persson surfaces, a family of Galois covers of the projective plane, and proves a Torelli-type theorem relating these surfaces to their Hodge structures.

Contribution

It explicitly describes the KSBA stable degenerations of Persson surfaces and establishes a generic global Torelli theorem for these surfaces.

Findings

The moduli stack of Persson surfaces is smooth after compactification.

Stable degenerations are described via weighted hyperplane arrangements.

A Torelli-type theorem recovers generic Persson surfaces from Hodge structures.

Abstract

We study a family of canonically polarized surfaces introduced by Persson, which arise as Galois $G=(\mathbb{Z}/2\mathbb{Z})^4$-covers of $\mathbf{P}^2$ branched along eight general lines. For this family, we construct the compactified moduli space and explicitly describe the stable degenerations in the sense of Koll\'ar, Shepherd-Barron, and Alexeev (KSBA) via stable pairs of weighted hyperplane arrangements. By computing the $\mathbb{Q}$-Gorenstein obstructions and using the KSBA wall crossings, we show that the resulting compactified moduli stack is smooth. Furthermore, we establish a generic global Torelli type result: up to two possibilities, a generic smooth Persson surface can be recovered from the Hodge structure on the anti-invariant part of the second cohomology of its \'etale double cover, together with the associated $\widetilde{G}=(\mathbb{Z}/2\mathbb{Z})^5$-action.