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

We classified prime Q-Fano 3-folds X with only 1/2(1, 1,1)-singularities and with h(0)(-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. In this paper, revealing the geometries behind the Sarkisov link for X in one of 5 classes, we show that X can be embedded as a linear section into a bigger dimensional Q-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.