Sarkisov links for index 1 Fano 3‐folds in codimension 4

Abstract

We classify Sarkisov links from index 1 Fano 3-folds anticanonically embedded in codimension 4 that start from so-called Type I Tom centres. We apply this to compute the Picard rank of many such Fano 3-folds.

Nonsolidity of uniruled varieties

“…Consider the deformation family with ID #39660 in Tom format. This is 𝑋 ⊂ P P (2,2,3,5,5,7,12,17) with homogeneous coordinates 𝜉, 𝑦 4 , 𝑦 1 , 𝑥 2 , 𝑦 2 , 𝑦 3 , 𝑥 1 , 𝑠. By Lemma 3.4, we know that the weighted blowup of p s = (0 : .…”

“…Note that, if 𝑠𝑦 𝑖 for some 1 ≤ 𝑖 ≤ 4 has odd weight, the corresponding 𝑔 𝑖 does not contain any pure monomial in 𝜉, 𝑥 1 ; that is, wt(𝑦 𝑖 ) must be odd for it to happen. We briefly recall the notation of unprojection equations necessary to this proof; for the full details of the construction, we refer to [32,Section 5.3] and [12,Appendix]. To fix ideas, suppose that the matrix M is in Tom 1 format; the proof for the other Tom formats is analogous.…”

“…Since some of the polynomials 𝑞 𝑖 are quasi-linear in the 𝑦 𝑖 , the polynomials 𝑝 𝑖 having even degrees are multiplied by 1 at least twice in the Pfaffians of the matrices 𝑁 𝑗 (also note that the matrices we consider have always weights as in [12,Equation (4.2) and (4.3)]). Thus, at least three entries of each row of the 4×4 matrix Q contain a monomial purely in 𝜉, 𝑥 1 .…”

“…These formats, defined by specific constraints on the polynomial entries of the matrix M (cf [9, Definition 2.2]), are called Tom and Jerry formats. Accordingly, X is said to be either of Tom type or of Jerry type (cf [12,Definition 2.2]). Not all the formats are compatible with the double-cover construction, that is, not all formats descend to index 2.…”

“…It is interesting to observe that the behaviour of the restriction of 𝜏 0 as in Lemma 4.5 is not a feature of the codimension of X but rather of its Fano index. When the index is 1, the map 𝜏 restricts to a number of simultaneous Atyiah flops (see [12,Theorem 4.1] and [29]). On the other hand, for higher indices it is an isomorphism (cf [21, Theorem 2.5.6]).…”