Moduli of double covers and degree one del Pezzo surfaces

TL;DR

This paper studies the moduli space of degree one del Pezzo surfaces, showing that their boundary can be explicitly described as double covers of degenerations of the quadric cone, extending known geometric descriptions.

Contribution

It provides an explicit classification of the boundary of the moduli space of degree one del Pezzo surfaces as double covers of degenerations of the quadric cone.

Findings

Boundary of moduli space characterized as double covers

Explicit classification of degenerations of the quadric cone

Extension of geometric description to boundary cases

Abstract

Given a degree one del Pezzo surface with canonical singularities, the linear series generated by twice the anti-canonical divisor exhibits the surface as the double cover of the quadric cone branched along a sextic curve. It is natural to ask if this description extends to the boundary of a compactification of the moduli space of degree one del Pezzo surfaces. The goal of this paper is to show that this is indeed the case. In particular, we give an explicit classification of the boundary of the moduli space of anti-canonically polarized broken del Pezzo surfaces of degree one as double covers of degenerations of the quadric cone.