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

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 Kollár, 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 étale double cover, together with the associated $\tilde{G}=(\mathbb{Z}/2\mathbb{Z}{)}^5$-action.

37 pages, 8 figures. Comments are welcome