On base point freeness in positive characteristic

By Paolo Cascini, Hiromu Tanaka. 2013

86 citations recorded by the source corpus.

We prove that if $(X,A+B)$ is a pair defined over an algebraically closed field of positive characteristic such that $(X,B)$ is strongly $F$-regular, $A$ is ample and $K_X+A+B$ is strictly nef, then $K_X+A+B$ is ample. Similarly, we prove that for a log pair $(X,A+B)$ with $A$ being ample and $B$ effective, $K_X+A+B$ is big if it is nef and of maximal nef dimension. As an application, we establish a rationality theorem for the nef threshold and various results towards the minimal model program in dimension three in positive characteristic.

Subfields: Geometry and Topology, History, Sociology and Political Science

View Paper: https://www.semanticscholar.org/paper/15bb8c8959c29aecaef5c0bb2153019cb7967392

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In Geometry and Topology

1. C. Birkar
2. Christopher D. Hacon
3. Hiromu Tanaka
4. J. Waldron
5. J. Witaszek
6. O. Fujino
7. Chi Li
8. Paolo Cascini

In History

1. C. Birkar
2. J. Waldron
3. Hiromu Tanaka

In Sociology and Political Science

1. C. Birkar
2. J. Waldron
3. Hiromu Tanaka

Citations in the corpus

• The augmented base locus of real divisors over arbitrary fields (2013, methodology): “rs. Moreover, suppose A is nef and big and L = K X +B +A is nef. Theorem 6.1. Assume Ld = 0. Then B +(L) = X is covered by rational curves C with L ·C = 0. The theorem was proved by Cascini-Tanaka-Xu [7] and independently by McKernan, when k has positive characteristic. A short proof of this in any ”
• On the rationality of Kawamata log terminal singularities in positive characteristic (2017, background): “In spite of this, it is known that the klt MMP is valid in dimension two [Tan14] and dimension three in characteristic p > 5 [HX15, CTX15, Bir16, BW17].”
• Existence of Mori fibre spaces for 3-folds in char $p$ (2014, methodology): “unique. When we speak of running such an LMMP we make sure that all the necessary ingredients exist, eg the extremal rays, the contractions of the rays, etc. 2.9. Nef reduction maps. Theorem 2.10 ([1][7]). Let Xbe a normal projective variety over an uncountable k, and La nef R-divisor on X. Then the”
• Existence of flips and minimal models for 3-folds in char p (2013, methodology): “ (X,B +A) is klt as well (9.2). By Theorem 1.2, (X,B + A) has a log minimal model (Y,BY + AY ). Since KY + BY + AY is not big, KY + BY + AY is numerically trivial on some covering family of curves by [9](see also 1.11 below). Again by [9], there is a nef reduction map Y 99K T for KY +BY +AY which is”
• Minimal model program for excellent surfaces (2016, methodology): “from Lemma 3.5.MINIMAL MODEL PROGRAMME 15 In the following proposition, we use notion of F-singularities. For definitions and basic properties, we refer to [CTX15, Section 2.3] (cf. [Sch09]). Although [CTX15] works over an algebraically closed field of positive characteristic, the same argument works ”