Looijenga's Conjecture via Integral-affine Geometry

TL;DR

This paper offers an alternative proof of Looijenga's conjecture on cusp singularities' smoothability, utilizing combinatorial criteria from Friedman and Miranda, contrasting with previous mirror symmetry approaches.

Contribution

It provides a new proof of Looijenga's conjecture using combinatorial methods instead of mirror symmetry techniques.

Findings

Established a combinatorial criterion for cusp singularity smoothability.

Proved the equivalence between the criterion and Looijenga's conjecture.

Enhanced understanding of cusp singularities via integral-affine geometry.

Abstract

A cusp singularity is an elliptic surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In 1981, Looijenga proved that whenever a cusp singularity is smoothable, the minimal resolution of the dual cusp is an anticanonical divisor of some smooth rational surface. He conjectured the converse. Recent work of Gross, Hacking, and Keel has proven Looijenga's conjecture using methods from mirror symmetry. This paper provides an alternative proof of Looijenga's conjecture based on a combinatorial criterion for smoothability given by Friedman and Miranda in 1983.