Mori dream spaces and birational rigidity of Fano 3-folds

Abstract

Abstract. We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove birational rigidity statements. To illustrate, we complete the birational rigidity results of Okada for Fano complete intersection 3-folds in singular weighted projective spaces.

Birationally rigid Pfaffian Fano 3-folds

“…Theorem 1.1, in particular, generalises a result of Iskovskikh and Pukhlikov that shows a general smooth complete intersection of a conic and a cubic in P 5 is birationally rigid, see [13] and [22, chapter 2]. Theorem 1.1 has been generalised for quasi-smooth models (without the generality conditions) by Ahmadinezhad and Zucconi [1].…”

“…and the set in the right-hand side of the above equation is finite (for any α, β, γ ∈ C). This shows that {x, y, z 0 } isolates p. We see ord E (x, y, z 0 ) ≥ 1 5 (1,4,5) so that L = B is nef by Lemma 3.3. We compute…”

“…Therefore, p is not a maximal centre by Lemma 3.1. (1,1,4). After replacing y 0 , y 1 , we may assume p = p y 1 .…”

“…MoreoverẐ is obtained as the Kawamata blowup of Z at the 1 5 (1, 1, 4) point q := ρ(ϕ −1 (q)). Since the point ϕ −1 (q) ∈ Y is the unique singular point of 1 5 (1, 1, 4), the pointq ∈ Z is the unique point of type 1 5 (1,1,4). Hence θ fixesq.…”

On Mori chamber and stable base locus decompositions

“…Indeed, if there exists a Mori dream space Z of Picard rank two admitting a dominant morphism Z Ñ X then such a factorization is determined by a so called 2-ray game on Z, and such a 2-ray game is in turn determined by the Mori chamber decomposition of EffpZq. We refer to [Cor95], [HM13], [AZ16], [Ahm17] for details on this topic and explicit examples.…”