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

Hiromichi Takagi

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arXiv · 2210.16008

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

Hiromichi Takagi

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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.

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Hiromichi Takagi. 2022-10-28. On Classification of $\mathbb{Q}$-Fano 3-folds of Gorenstein index 2. III. https://arxiv.org/abs/2210.16008

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