Classification of ℚ-Fano 3-folds of Gorenstein index 2 via key varieties constructed from projective bundles

Abstract

We classified prime [Formula: see text]-Fano [Formula: see text]-folds [Formula: see text] with only [Formula: see text]-singularities and with [Formula: see text] a long time ago. The classification was undertaken by blowing up each [Formula: see text] at one [Formula: see text]-singularity and constructing a Sarkisov link. In this paper, revealing the geometries behind the Sarkisov link for [Formula: see text] in one of 5 classes, we show that [Formula: see text] can be embedded as a linear section into a bigger dimensional [Formula: see text]-Fano variety called a key variety. The key variety is constructed by extending partially the (modified) Sarkisov link in higher dimension, and turns out to be birational to a projective bundle over a certain Fano manifold.