Stable pair compactifications of the moduli space of degree one del pezzo surfaces via elliptic fibrations

TL;DR

This paper constructs a stable pair compactification of the moduli space of degree one del Pezzo surfaces using elliptic fibrations, linking classical geometry with modern moduli theory.

Contribution

It introduces a novel compactification approach for the moduli space of degree one del Pezzo surfaces via elliptic fibrations and stable pairs.

Findings

Established a stable pair compactification of the moduli space

Connected classical Cayley-Bacharach theorem with modern moduli theory

Provided a new geometric perspective on degree one del Pezzo surfaces

Abstract

A degree one del Pezzo surface is the blowup of P^2 at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we construct a stable pair compactification of the moduli space of anti-canonically polarized degree one del Pezzo surfaces.