On Classification of ℚ-Fano 3-folds of Gorenstein index 2. III

2022/10/28 by Hiromichi Takagi, Takagi, Hiromichi
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2210.16008

openalex publication_date 2022/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classified prime ℚ-Fano 3-folds X with only 1/2(1,1,1)-singularities and with h⁰(-K_X)≥ 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 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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