The augmented base locus of real divisors over arbitrary fields

We show that the augmented base locus coincides with the exceptional locus (i.e. null locus) for any nef $\mathbb{R}$-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness criterion in terms of the augmented base locus generalizing a result of Keel. We also study nef divisors with positive top intersection number, and discuss some problems related to augmented base loci of log divisors.Leverhulme TrustThis is the author accepted manuscript. The final version is available from Springer via http://dx.doi.org/10.1007/s00208-016-1441-

Birational automorphism groups of projective varieties of Picard number two

We slightly extend a result of Oguiso on birational or automorphism groups (resp. of Lazi\'c - Peternell on Morrison-Kawamata cone conjecture) from Calabi-Yau manifolds of Picard number two to arbitrary singular varieties X (resp. to klt Calabi-Yau pairs in broad sense) of Picard number two. When X has only klt singularities and is not a complex torus, we show that either Aut(X) is almost cyclic, or it has only finitely many connected components.Comment: title slightly changed to this; some proof simplified; submitted to the Proceedings of Groups of Automorphisms in Birational and Affine Geometry, 28 October - 3 November 2012, C.I.R.M., Trento, Ital

Existence of Mori fibre spaces for 3-folds in char p

We prove the following results for projective klt pairs of dimension 3 over an algebraically closed field of characteristic $p$>5: the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, existence of Mori fibre spaces, etc.This work was partially supported by a grant of the Leverhulme Trust. Part of this work was done when the first author visited National Taiwan University in August– September 2014 with the support of the Mathematics Division (Taipei Office) of the National Center for Theoretical Sciences

Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

For every smooth complex projective variety W of dimension d and nonnegative Kodaira dimension, we show the existence of a universal constant m depending only on d and two natural invariants of the very general fibres of an Iitaka fibration of W such that the pluricanonical system | mKW| defines an Iitaka fibration. This is a consequence of a more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc.The first author was partially supported by a grant of the Leverhulme Trust. Part of this work was done when the first author visited National University of Singapore in April 2014. Part of this work was done when the first author visited National Taiwan University in August-September 2014 with the support of the Mathematics Division (Taipei Office) of the National Center for Theoretical Sciences. The visit was arranged by Jungkai A. Chen. He wishes to thank them all. The second author was partially supported by an ARF of National University of Singapore.http://dx.doi.org/10.1007/s10240-016-0080-

Algebraic varieties with automorphism groups of maximal rank

We confirm, to some extent, the belief that a projective variety X has the largest number (relative to the dimension of X) of independent commuting automorphisms of positive entropy only when X is birational to a complex torus or a quotient of a torus. We also include an addendum to an early paper though it is not used in the present paper.Comment: Mathematische Annalen (to appear

Weakly--exceptional quotient singularities

A singularity is said to be weakly--exceptional if it has a unique purely log terminal blow up. In dimension $2$, V. Shokurov proved that weakly--exceptional quotient singularities are exactly those of types $D_{n}$, $E_{6}$, $E_{7}$, $E_{8}$. This paper classifies the weakly--exceptional quotient singularities in dimensions $3$ and $4$

Public support in England for a total ban on the sale of tobacco products

Background This study aimed to determine the level of support for a sales ban on tobacco in England to provide a benchmark against which any changes over time can be assessed.Methods 8735 people from England who participated in one of five monthly cross-sectional household surveys in 2008 were asked to indicate whether they would support the statement that 'the government should work towards banning the sale of tobacco completely within the next 10 years'. In addition, sociodemographic and smoking characteristics were assessed.Results A substantial proportion of the total sample (44.5%; 95% CI 43.5% to 45.6%) would support a move towards a complete ban. While never smokers (OR 2.02; 95% CI 1.82 to 2.25) and ex-smokers (OR 1.41; 95% CI 1.21 to 1.65) were more likely to support this idea, even among current smokers, a third would favour moving towards a sales ban of tobacco products. Adjusting for other background characteristics, younger, female participants, those living in London and those from lower socioeconomic groups were most likely to support a ban. Among smokers, a higher cigarette consumption, smoking enjoyment and contentment with being a smoker were associated with opposition to a ban, while feeling uncomfortable being a smoker, wanting to be a non-smoker and being worried about future health consequences of smoking were associated with support for a ban.Conclusion Support for movement towards a ban on the sale of tobacco is higher than might be imagined. It is conceivable that as smoking prevalence falls further and smoking becomes more socially unacceptable, support might grow to a point where such a policy could become feasible

A travel guide to the canonical bundle formula

We survey known results on the canonical bundle formula and its applications in algebraic geometry.Comment: 17 pages, to appear in the Proceedings of the conference Birational Geometry and Moduli Space

Differential Forms on Log Canonical Spaces

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting. Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.Comment: 72 pages, 6 figures. A shortened version of this paper has appeared in Publications math\'ematiques de l'IH\'ES. The final publication is available at http://www.springerlink.co