Constructing and characterizing prime $\mathbb{Q}$-Fano threefolds of genus one and with six $1/2(1,1,1)$-singularites via key varieties

Abstract

We consider the classification problem of prime $\mathbb{Q}$-Fano 3-folds with at most $1/2(1,1,1)$-singularities, which was initiated in [Taka2]. We construct two distinct classes of such 3-folds with genus one and six $1/2(1,1,1)$-singularities, each equipped with a prescribed Sarkisov link. Our method involves constructing certain higher-dimensional $\mathbb{Q}$-Fano varieties $Σ$, referred to as key varieties, by extending the Sarkisov links to higher dimensions. We prove that each such 3-fold $X$ arises as a linear section of the corresponding key variety $Σ$, and conversely, any general linear section of $Σ$ yields such an $X$. Various geometric properties of the key varieties $Σ$ are also investigated and clarified.

Theorems & Definitions (154)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof