Algebraic tori in the complement of quartic surfaces

Abstract

Let $B\subset {\mathbb{P}}^3$ be an slc quartic surface. The existence of an embedding ${\mathbb{G}}_m^3\hookrightarrow {\mathbb{P}}^3\setminus B$ implies that $B$ has coregularity zero. In this article, we initiate the classification of coregularity zero slc quartic surfaces $B\subset {\mathbb{P}}^3$ for which ${\mathbb{P}}^3\setminus B$ contains an algebraic torus ${\mathbb{G}}_m^3$. Equivalently, the classification of cluster type pairs $({\mathbb{P}}^3,B)$. Along the way, we give criteria for a log Calabi--Yau pair $(X,B)$ over a toric variety $T$ to be of cluster type.

24 pages