Finite generation and geography of models

Abstract

There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which are finitely generated, we determine precisely when we can run the MMP, and we show why finite generation alone is not sufficient to make the MMP work.

Geometric aspects of Newton–Okounkov bodies

“…The decomposition obtained this way is quite close to the one discussed in [HK00] or [KKL17], in fact, when the surface under consideration is a Mori dream space, the open parts of the chambers in the respective decompositions agree. However, it is a significant difference that here we do not rely on any kind of finite generation hypothesis, the decomposition of the cone of big divisors that we present always exists.…”

“…The main tool to compute Newton-Okounkov bodies of divisors on surfaces is Zariski decomposition, whose variation inside the big cone determines their shapes. This phenomenon is strongly related to decompositions of big or effective cones of varieties (see [HK00,KKL17]). In fact, along with toric varieties, this is the class of non-trivial examples that is the best understood.…”

On K-stability and the volume functions of ℚ-Fano varieties: Table 1.

“…In this section, we recall the theory of "geography of models" introduced in [Sho96, Section 6]. For the notation in this section, we refer the readers to [KKL12].…”

The Sarkisov program for Mori fibred Calabi-Yau pairs

“…For the proof see for example [7,Theorem 4.5]. Note that the assumptions readily imply that Supp R contains big divisors.…”

“…Note that K Yi−1 + G Yi−1 , K Yi−1 + G Yi−1 + A i−1 generic implies K Yi + G Yi , K Yi + G Yi + A i generic. 7 We take p i : X i → S i to be the end product of the MMP for K Yi + G Yi with scaling by A i . It follows from lemma 32, applied to Y i−1 and the divisors A i−1 , A i−1 , that the induced map ϕ i : X i−1 X i is a composition of Sarkisov links under Y i−1 and hence, since Y Y i−1 is an isomorphism in codimension one, under Y .…”