The KSBA compactification of the moduli space of $D_{1,6}$-polarized Enriques surfaces

TL;DR

This paper constructs a KSBA compactification of a specific family of Enriques surfaces, revealing its structure as a toric variety and relating it to other known compactifications.

Contribution

It introduces a KSBA compactification for the moduli space of certain Enriques surfaces and connects it with toric, Baily-Borel, and Looijenga's semitoric compactifications.

Findings

The compactification is isomorphic to a toric variety associated with the secondary polytope of a cube.

Part of the boundary exhibits toroidal behavior, while another part matches the Baily-Borel compactification.

The study links the stable pair compactification with various known compactification types.

Abstract

We describe a compactification by stable pairs (also known as KSBA compactification) of the $4$-dimensional family of Enriques surfaces which arise as the $\mathbb{Z}_2^2$-covers of the blow up of $\mathbb{P}^2$ at three general points branched along a configuration of three pairs of lines. Up to a finite group action, we show that this compactification is isomorphic to the toric variety associated to the secondary polytope of the unit cube. We relate the KSBA compactification considered to the Baily-Borel compactification of the same family of Enriques surfaces. Part of the KSBA boundary has a toroidal behavior, another part is isomorphic to the Baily-Borel compactification, and what remains is a mixture of these two. We relate the stable pair compactification studied here with Looijenga's semitoric compactifications.