Lecture courses

Birational Geometry in characteristic p - James M^cKernan (MIT)

Abstract: We will discuss the Minimal Model Program in finite characteristic, with emphasis on the Cone Theorem and, possibly, basepoint freeness following Sean Keel's paper.

Singularities as measured by Frobenius and applications - Karl Schwede (Penn State)

1. Introduction to F-singularities I
   Topics covered: Test ideals, F-pure and F-regular rings and relation to characteristic zero.
2. Frobenius splittings and geometric applications
   Topics covered: Globally F-split and F-regular varieties, divisors vs maps, vanishing theorems. (This is largely motivational)
3. Introduction to F-singularities II
   Topics covered: More on test ideals, divisors vs maps, F-pure centers, F-adjunction.
4. Lifting and producing sections by Frobenius
   Topics covered: Global generation statements for test ideals, pulling back sections via adjunction.
5. F-Seshadri constants and related constructions
   Topics covered: F-Seshadri constants, possibly some discussion of Hilbert-Kunz multiplicity and F-signature.
6. Global applications of F-singularities Topics covered: Various recent global applications of F-singularities.

Invited talks

On varieties of globally F-regular type. - Yoshinori Gongyo (Imperial)

Abstract: I will talk about some topics of varieties of globally F-regular type that is defined by using Frobenius map.

Products of involutions in the Cremona group. - Nick Shepherd-Barron (Cambridge)

Abstract: Birational automorphisms of the plane can have complicated dynamics. The aim is to estimate, or even calculate, some of these dynamical quantities when such an automorphism is given in explicit algebraic terms. This is motivated by problems in symmetric key cryptography, but is also heavily influenced by work of Cantat and Lamy, McMullen and Gromov.

Simultaneous resolution of simple singularities without simple groups. - Nick Shepherd-Barron (Cambridge)

Abstract: In good characteristics (which excludes characteristics 2,3 and 5 in the E_8 case) deformations of du Val singularities can be embedded into the adjoint quotient of the corresponding simply connected simple algebraic group. The existence of simultaneous resolutions after a base change by the corresponding Weyl group, and the classification of 3-fold flops, follows at once. In this talk we show that the Weyl group picture remains valid in all characteristics, although the relations with algebraic groups break down.

The trace map of Frobenius and extending sections. - Hiromu Tanaka (Kyoto)

Abstract: We consider problems on extending sections for positive characteristic varieties. However, there exist some bad examples in the case where the dimension is two or three.

Contributed talks

Elementary counter-examples to Kodaira vanishing in prime characteristic. - Kuzma Khrabrov (Warwick/Higher School of Economics, Moscow)

Abstract: In my talk I am going to start following ideas of the first set of counterexamples to Kodaira vanishing by M. Raynaud. I will proceed constructing elementary examples from the paper by N. Lauritzen and A. P. Rao. The latter examples are connected with incidence correspondence of affine subspaces in some vector space and natural vector bundles arising on such varieties. We will construct a smooth projective variety and a very ample sheaf on it, which violates Kodaira vanishing.

On the upper semi-continuity of the HSL numbers. - Serena Murru (Sheffield)

Abstract: Let B be an affine Cohen-Macaulay algebra over a field of positive characteristic.
For every prime ideal p of B, let B_p be the localisation and completion of B with respect to p and let H_p be the top-local cohomology of B_p with support pB_p. Each such H_p is an Artinian module endowed with a natural Frobenius map F and if Nil(H_p) denotes the set of all elements in H_p killed by some power of F then a theorem by Hartshorne-Speiser and Lyubeznik shows that there exists an e>0 such that F^e Nil(H_p)=0. The smallest such e is the HSL-number of H_p which we denote HSL(H_p).

Faithful action on the spaces of holomorphic poly-differentials on curves - Joe Tait (Southampton)

Abstract: Given a finite group G acting on a projective curve X over an algebraically closed field k we consider when the induced action on the space of holomorphic poly-differentials is faithful. We answer this question fully, without restricting the characteristic of k. We then look at hyperelliptic curves as an explicit example. .

Poster Session

Non-isotrivial fibrations of irregular surfaces. - Victor Gonzalez Alonso (UPC)
Free curves on varieties. - Frank Gounelas (Heidelberg)
Bogomolov-Sommese vanishing on log canonical pairs. - Patrick Graf(Freiburg)
Poincaré Lemmas on Singular Spaces. - Clemens Jörder (Freiburg)
Classifying bad log pairs: a tale of cats and tigers. - Jesus Martinez-Garcia (Edinburgh)
Minimal, Stable and Canonical Models of Arithmetic Surfaces. - Thomas Oliver (Nottingham)
Weakly exceptional quotient singularities. - Dmitrijs Sakovics (Edinburgh)
Deformation of weak Fano manifolds. - Taro Sano (Warwick)
Bounding the Genus of Curves Lying on Complete Intersection Threefolds. - Rebecca Tramel (Edinburgh)
Weil divisors on normal varieties. - Stefano Urbinati (Warsaw)
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