ADE surfaces and their moduli

TL;DR

This paper introduces ADE surfaces linked to root lattices, constructs their moduli space compactifications as quotients of projective varieties, and applies these to rational elliptic surfaces, extending known moduli space theories.

Contribution

It defines ADE surfaces, constructs their moduli space compactifications via Coxeter fans, and provides a geometric compactification of rational elliptic surface moduli.

Findings

Moduli spaces are quotients of projective varieties for Coxeter fans.

Constructed modular families extend to stable pairs over compactifications.

Achieved a finite quotient of a projective toric variety as a moduli space.

Abstract

We define a class of surfaces corresponding to the ADE root lattices and construct compactifications of their moduli spaces as quotients of projective varieties for Coxeter fans, generalizing Losev-Manin spaces of curves. We exhibit modular families over these moduli spaces, which extend to families of stable pairs over the compactifications. One simple application is a geometric compactification of the moduli of rational elliptic surfaces that is a finite quotient of a projective toric variety.