Algebraic tori in the complement of quartic surfaces

TL;DR

This paper classifies certain quartic surfaces in projective 3-space whose complements contain algebraic tori, linking geometric properties with cluster structures and coregularity zero conditions.

Contribution

It initiates the classification of coregularity zero slc quartic surfaces with algebraic torus embeddings in their complements, connecting cluster types with geometric and singularity conditions.

Findings

Classification of coregularity zero slc quartic surfaces with torus embeddings

Criteria for log Calabi--Yau pairs to be of cluster type

Establishment of conditions linking surface properties to algebraic tori

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.