Quartic surfaces up to volume preserving equivalence

Abstract

We study log Calabi--Yau pairs of the form $(\mathbb{P}^3,\Delta)$, where $\Delta$ is a quartic surface, and classify all such pairs of coregularity less than or equal to one, up to volume preserving equivalence. In particular, if $(\mathbb{P}^3,\Delta)$ is a maximal log Calabi--Yau pair then we show that it has a toric model.