On Classification of $\mathbb{Q}$-Fano 3-folds of Gorenstein index 2. III

TL;DR

This paper explores the geometric structures of certain $Q$-Fano 3-folds with specific singularities, revealing that they can be embedded into higher-dimensional key varieties through extended Sarkisov links.

Contribution

It demonstrates that $Q$-Fano 3-folds in five classes can be embedded as linear sections into larger key varieties, extending Sarkisov links to higher dimensions.

Findings

Classification of $Q$-Fano 3-folds with specific singularities.

Embedding of these 3-folds into higher-dimensional key varieties.

Extension of Sarkisov links in higher dimensions.

Abstract

We classified prime $\mathbb{Q}$-Fano $3$-folds $X$ with only $1/2(1,1,1)$-singularities and with $h^{0}(-K_{X})\geq 4$ a long time ago. The classification was undertaken by blowing up each $X$ at one $1/2(1,1,1)$-singularity and constructing a Sarkisov link. The purpose of this paper is to reveal the geometries behind the Sarkisov links for $X$ in 5 classes. The main result asserts that any $X$ in the 5 classes can be embedded as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called key varieties, where the key varieties are constructed by extending partially the Sarkisov link in higher dimensions.