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

TL;DR

This paper classifies prime $Q$-Fano threefolds with specific singularities and genus, constructing key higher-dimensional varieties called key varieties, and shows these threefolds can be obtained as linear sections of these key varieties.

Contribution

It introduces the concept of key varieties to construct and classify prime $Q$-Fano threefolds with particular singularities and genus, extending Sarkisov links to higher dimensions.

Findings

Constructed two classes of prime $Q$-Fano threefolds with six $1/2(1,1,1)$-singularities.

Proved each threefold arises as a linear section of a higher-dimensional key variety.

Analyzed geometric properties of the key varieties $Sigma$.

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 $\Sigma$, 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 $\Sigma$, and conversely, any general linear section of $\Sigma$ yields such an $X$. Various geometric properties of the key varieties $\Sigma$ are also investigated and clarified.