Key variety construction of Sarkisov links for prime $\mathbb{Q}$-Fano threefolds of codimension four associated to Type ${\rm II}_{2}$ projections

TL;DR

This paper explicitly constructs Sarkisov links for certain prime $Q$-Fano threefolds of codimension four, revealing their end structures as del Pezzo fibrations or divisorial contractions.

Contribution

It provides explicit constructions of Sarkisov links for seven families of prime $Q$-Fano threefolds, detailing their end structures and associated geometric transformations.

Findings

Sarkisov links end with del Pezzo surface fibrations or divisorial contractions.

Explicit descriptions of Sarkisov links for seven families of threefolds.

Connections between weighted projectivizations and Sarkisov link structures.

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 $\widehat{f}\colon\widehat{X}\to X$ be the unique divisorial extraction at one specified singularity of maximal index. In this paper, we explicitly construct the Sarkisov link starting from $\widehat{f}$ 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.