On the log discrepancies in toric Mori contractions

Abstract

It was conjectured by M c Kernan and Shokurov that for all Mori contractions from X to Y of given dimensions, for any positive ε there is a positive δ such that if X is ε-log terminal, then Y is δ-log terminal. We prove this conjecture in the toric case and discuss the dependence of δ on ε, which seems mysterious.

Birational Geometry of Algebraic Varieties

“…On the other hand, independently, Shokurov proposed a more general problem which generalised both parts of Mori and Prokhorov result. M c Kernan's conjecture is known in the toric case [3]. Shokurov's conjecture is known when dim X − dim Z ≤ 1 [9], in particular for surfaces, and open in higher dimension but we have the following general result [9].…”

“…Sketch of proof of BAB. (Theorem 3.7) First applying [26,Theorem 1,3] it is enough to show that K X has a klt strong m-complement for some bounded number m ∈ N. Running an MMP on −K X and replacing X with the resulting model we can assume B = 0. By Theorem 3.3, we know that we have an lc strong n-complement K X + B + .…”

The rationality problem for conic bundles

“…This conjecture in the toric case was proved by Alexeev and Borisov [AB14]. Partial results in general case were obtained in [Bir16b] and related to log adjunction (cf.…”

Shokurov's conjecture on conic bundles with canonical singularities

“…(1) The formulation of Conjecture 1.5 here is stronger than that in the previous literature [AB14,Bir16], where a stronger assumption (2') that "(X, B) is an ǫ-lc pair" is required instead of assumption (2), and δ depends on dim X and ǫ instead of just dim X − dim Z and ǫ. In our formulation, B can be non-effective and (X, B) can have non-klt centers over Z \ z.…”