Key variety construction of Sarkisov links for prime $\mathbb{Q}$-Fano threefolds of codimension four associated to Type ${\mathrm{II}}_2$ projections

Abstract

In our paper [Tak6], we constructed eight families of quasi-smooth prime $\mathbb{Q}$-Fano threefolds, anticanonically embedded in codimension four, using weighted projectivizations of the $14$-dimensional affine variety ${\Pi }_{\mathbb{A}}^{14}$or its cone. Let $\hat{f}:\hat{X}\to X$ be the unique divisorial extraction at one specified singularity of maximal index. In this paper, we explicitly construct the Sarkisov link starting from $\hat{f}$ for $X$ belonging to seven of these families. This is achieved by using the Sarkisov link associated with the weighted projectivization of ${\Pi }_{\mathbb{A}}^{14}$ or its cone corresponding to $X$. As a consequence, we show that the Sarkisov link ends with either a fibration whose general fiber is a del Pezzo surface of degree one or a divisorial contraction of type $(2,1)$ to weighted complete intersections of codimension at most two. We also provide more detailed descriptions of these Sarkisov links.