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.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• Definition 2.7