ICMS

Jan 10, 2011 - Jan 14, 2011

ICMS, 15 South College Street Edinburgh

Organisers

Name	Institution
Batyrev, Victor	Universität Tübingen
Skorobogatov, Alexei N	Imperial College London

The aim of the workshop is to bring together those who work with torsors from different perspectives (geometric invariant theory, computational algebraic geometry, representation theory, non-abelian cohomology) and those who apply torsors in arithmetic geometry. 

Short report

The idea of the meeting was to bring together mathematicians who use torsors in their research. Two mini-courses presented recent achievements on the borderline between geometric invariant theory and representation theory. One morning session was devoted to new applications of étale homotopy; arithmetic applications of torsors featured in a number of other talks. On the whole, talks were accessible to a heterogeneous audience of experts, postdocs and PhD students coming from different backgrounds (algebra, algebraic geometry and arithmetic geometry). We had a good mix of speakers: well established influential researchers as well as brilliant younger people. Much discussion went on between the talks. One goal has already been achieved: mathematicians working in areas that are related but which had limited contact have become aware of the existing methods and techniques. We are confident that this cross-fertilisation of ideas will continue. There was sufficient enthusiasm to publish a volume of conference proceedings featuring lecture notes of the two mini-courses.

The full report can be found here.

 

Arrangements

Participation
Participation is by invitation only. The workshop will begin on Monday morning 10 January 2011 and finish on Friday afternoon 14 January 2011.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and, if so, how to apply for one. If you do require a visa, ICMS can (on request) provide a signed invitation letter.

Venue
The workshop will take place at 15 South College Street, ICMS headquarters. South College Street is a small side street of Nicolson Street and it is next to Old College which is a local landmark. If you need to ask for directions, most people should know these places.

Wireless Access
ICMS at 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network.  For those without laptops, there are a number of computers available for you to check your emails. Delegates are welcome to make use of the public areas at 15 South College Street throughout the week.

Accommodation
Delegates who wish to have accommodation booked for them are asked to tick the appropriate box on the registration form. Accommodation is block booked for single occupancy. Delegates who plan to travel with partners and/or family should contact ICMS to discuss this. All booked accommodation is located in the Pollock Halls area (a map will be available here shortly). When arriving to the Pollock Halls campus please go to the Reception Centre to check in or for further instructions (during some hours check-in may be available in the respective accommodations). The Reception Centre is manned 24/7.

Wednesday 12 January - afternoon activity
We are very pleased to be able to offer a very interesting guided tour of Edinburgh's Historic Vaults during the free afternoon on Wednesday 12 January. More details will follow.

Map
This map shows the location of the Pollock Halls area where the accommodation is located, and the location of the workshop venue (ICMS, 15 South College Street).

Insurance
All delegates must ensure their travel insurance cover is adequate. ICMS cannot be held responsible for costs incurred in case of flight delays or similar.  

Programme

Programme published 14 December 2010. Last amendment 13 January 2011.

 

Monday 10 January

09.00 - 09.55	Registration and coffee/tea
09.55 - 10.00	Welcome & Health and Safety information
10.00 - 10.50	Victor Batyrev (Universität Tübingen) Distribution of rational points and Mori program
10.50 - 11.20	Coffee and tea served in the Chapterhouse
11.20 - 12.20	Jürgen Hausen (Universität Tübingen) Cox rings and geometric invariant theory (part I) Link to slides
12.20 - 14.30	Lunch served in the Chapterhouse
14.30 - 15.30	Vera Serganova (University of California at Berkley) Representation theory, torsors and del Pezzo surfaces (part I) Link to slides
15.30 - 16.00	Coffee and tea served in the Chapterhouse
16.00 - 16.50	Jacob Stix (Universität Heidelberg) On the section conjecture over finite fields
17.00 - 17.50	Felipe Voloch (University of Texas at Austin) Finite descent obstructions for integral points on curves Link to slides
18.00 - 19.30	Wine reception

Tuesday 11 January

09.00 - 10.00	Jürgen Hausen (Universität Tübingen) Cox rings and geometric invariant theory (part II) Link to slides
10.10 - 11.10	Vera Serganova (University of California at Berkley) Representation theory, torsors and del Pezzo surfaces (part II) Link to slides
11.10 - 11.40	Coffee and tea served in the Chapterhouse
11.40 - 12.30	Phillipe Gille (Ecole Normale Supérieure) Torsors and classification of multiloop Lie algebras
12.30 - 15.00	Lunch
15.00 - 15.50	Mathieu Florence (Université Paris 6) Equivariant geometry of Grassmannians
15.50 - 16.20	Coffee and tea served in the Chapterhouse
16.20 - 17.10	Jenia Tevelev (University of Massachusetts at Amherst) Geometric tropicalization and Cox coordinates

Wednesday 12 January

09.00 - 09.50	Ambrus Pál (Imperial College London) Rational points and homotopy theory
10.00 - 10.50	Yonatan Harpaz (The Hebrew University in Jerusalem) Homotopy obstructions to integral points
10.50 - 11.20	Coffee and tea served in the Chapterhouse
11.20 - 12.10	Tomer Shlank (The Hebrew University in Jerusalem) The Brauer- Manin obstruction applied to ramified covers
12.10- 13.45	Lunch
13.45 - 15.00	Extracurricular activity: Guided tour of the Historic Vaults (more details to follow)

Thursday 13 January

09.00 - 10.00	Jürgen Hausen (Universität Tübingen) Cox rings and geometric invariant theory (part III) Link to slides
10.10 - 11.10	Vera Serganova (University of California at Berkley) Representation theory, torsors and del Pezzo surfaces (part III) Link to slides
11.10 - 11.40	Coffee and tea served in the Chapterhouse
11.40 - 12.30	Ulrich Derenthal (Ludwig-Maximilians-Universität München) Counting rational points via universal torsors
12.30 - 15.00	Lunch
15.00 - 15.50	Antonio Laface (Universidad de Concepción) Techniques for calculating Cox rings
15.50 - 16.20	Coffee and tea served in the Chapterhouse
16.20 - 17.10	Ivan Arzhantsev (Lomonosov Moscow State University) Two applications of Cox rings in affine geometry
19.00-	Workshop dinner at The Magnum Restaurant, 1 Albany Street

Friday 14 January

09.00 - 09.50	David Harari (Universite Paris-Sud) Descent theory for open varieties
10.00 - 10.50	Cyril Demarche (Université Paris 6) Strong approximation in homogeneous spaces over number fields
10.50 - 11.20	Coffee and tea served in the Chapterhouse
11.20 - 12.10	Mikhail Borovoi (Tel Aviv University) Extended Picard complex of a homogeneous space
12.20 - 13.10	Sergey Gayfullin (Moscow State University) Cox rings of S-varieties
	End of workshop

 

Presentations:

Presentation Details
Arzhantsev, Ivan
Two applications of Cox rings in affine geometry
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The aim of this talk is to describe the Cox rings of certain complex affine algebraic varieties and to apply this description to geometric problems. In the first part we study the Cox rings of Luna strata for linear actions of reductive algebraic groups. In particular, we calculate the divisor class group of the strata. The second part is based on a joint work with H.Flenner, S.Kaliman, F.Kutzschebauch and M.Zaidenberg. Let us say that an affine variety X is flexible if the tangent space of any smooth point on X is generated by sections of locally nilpotent vector fields. We prove that flexibility implies that the action of the group of special automorphisms on the smooth locus of X is infinitely transitive provided dim X is at least two. Using a description of the Cox ring of a normal affine SL(2)-embedding due to V.Batyrev and F.Haddad, we show that any such embedding is flexible.
Batyrev, Victor
Distribution of rational points and Mori program
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We give an overview of connections between the Mori program (via Cox rings) and the diophantine problems about the distribution of rational points of bounded height on algebraic varieties of negative Kodaira dimension.
Borovoi, Mikhail
Extended Picard complex of a homogeneous space
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Let k be an algebraically closed field of characteristic 0. Let G be a connected linear k-group acting on a smooth irreducible k-variety X. In works of Popov, Kraft, Knop, Vust, and others the equivariant Picard group Pic_G(X) of G-line-bundles on X was investigated. We introduce the extended equivariant Picard complex UPic_G(X) in degrees 0 and 1, whose first cohomology is Pic_G(X). In other words, we compute Pic_G(X) in terms of divisors and rational functions on X and X x G. This extended equivariant Picard complex UPic_G(X) is an equivariant version of the extended Picard complex UPic(X), introduced in our previous paper. When Pic(G)=0 and X is a homogeneous space of G with stabilizer H, we compute UPic(X) (up to an isomorphism in the derived category) in terms of the character groups of G and H. Using this result, we compute the "algebraic part" Br_a(X) of the Brauer group Br(X), when X is a homogeneous space (in particular, a torsor) defined over a number field. This is a joint work with the late Joost van Hamel.
Demarche, Cyril
Strong approximation in homogeneous spaces over number fields
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Given a quadratic (resp. hermitian) form f(x_1, ..., x_n) with coefficients in the ring of integers of a number field k and an integer a in this ring, one can prove that the existence of local integral solutions to the equation f(x_1,...,x_n)=a (at all places of k) is not sufficient (in general) for the existence of global integral solutions to this equation. We can introduce a natural obstruction called the Brauer-Manin obstruction to the integral Hasse principle, involving the Brauer group of the k-variety X defined by the equation. This variety X is a homogeneous space of the orthogonal (resp. symplectic) group. In this talk, we will explain the following recent result (and the role played by torsors in its proof) which extend earlier works by Kneser, Platonov, Colliot-Thélène and Xu, and Harari : the Brauer-Manin obstruction to the integral Hasse principle (resp. to strong approximation) for homogeneous spaces is the only one. This is a joint work with Mikhail Borovoi.
Derenthal, Ulrich
Counting rational points via universal torsors
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Fano varieties, for example cubic surfaces, often contain infinitely many rational points. A conjecture of Manin predicts their distribution precisely. One approach to this conjecture uses a parametrization of the rational points on the Fano variety by integral points on universal torsors. For certain singular cubic surfaces, this leads to a proof of Manin's conjecture.
Gayfullin, Sergey
Cox rings of S-varieties
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S-variety is an affine algebraic variety X with a locally-transitive action of a connected affine group G such that the stability subgroup of a generic point contains a maximal unipotent subgroup U of G. There is a bijection between S-varieties and finitely generated subsemigroups of the semigroup of dominant weights of G (Vinberg-Popov 1972). I can compute the Cox ring of any normal S-variety. The total coordinate space of the variety X is isomorphic to the product of a factorial S-variety Y (of the same group G) and an affine space. The semigroup corresponding to Y and the dimansion of the affine space can be obtained from the semigroup of X. This result allowes to compute the divisor class group of X.
Gille, Philippe
Torsors and classification of multiloop Lie algebras
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Our goal is to classify twisted toroidal Lie algebras namely Lie algebras which look like a simple complex Lie algebra g extended to a Laurent polynomial ring R. We shall discuss mainly the case of the Lie algebra g=sl(n) which relates to projective representations of finite abelian groups, quantum tori and Brauer groups of henselian fields. This is a report on joint work with Arturo Pianzola.
Harari, David
Descent theory for open varieties
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We extend descent theory of Colliot-Thelene and Sansuc to arbitrary smooth varieties, using Galois hypercohomology of complexes instead of cohomology of plain Galois modules. Various arithmetic applications are given (joint work with A.N. Skorobogatov).
Harpaz, Yonatan
Homotopy obstructions to integral points
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We describe a way to use a relative version of the etale homotopy type in order to construct obstructions to the existence of integral points and construct an example over a number ring where this obstruction is non-trivial although the integral Brauer Manin obstruction is trivial.
Laface, Antonio
Techniques for calculating Cox rings
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In this talk I will explain three techniques for determining a presentation of the Cox ring R(X) of an algebraic variety X/k. The first one, which focuses on the ideal of relations, is based on the calculation of Tor_1(R[X],k). The second one deals with cyclic coverings and their relation with Cox rings. Finally I will talk about hypersurfaces of varieties whose Cox ring is finitely generated.
Pál, Ambrus
Rational points and homotopy theory
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It is possible to formulate very general local-global principles using homotopy theories of algebraic varieties. In many cases these principles specialise to well-known obstructions or conjectures, in other cases they tell something new. In my talk I will try to give an overview of this subject.
Shlank, Tomer
The Brauer- Manin obstruction applied to ramified covers
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The Brauer-Manin obstruction is used to explain the failure of the local-global principle for algebraic varieties. In 1999 Skorobogatov gave the first example of a variety that does not satisfy the local-global principle which is not explained by the Brauer-Manin obstruction. He did so by applying the Brauer-Manin obstruction to '{e}tale covers of the variety, and thus defining a finer obstruction. In 2008 Poonen gave the first example of failure of the local-global principle which cannot be explained for by Skorobogatov's '{e}tale-Brauer obstruction. However, Poonen's construction was not accompanied by a definition of a new finer obstruction. In this paper I shall present a possible definition for such an obstruction by allowing to apply the Brauer-Manin obstruction to some ramified covers as well, and show that this new obstruction can explain Poonen counterexample in the case of a totally imaginary number field.
Stix, Jakob
On the section conjecture over finite fields
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Grothendieck's section conjecture in anabelian geometry suggests to use splittings of the fundamental group exact sequence to study rational points. Over number fields, splittings which locally come from rational points are related to adelic points which survive all finite torsor obstructions. We will discuss in the talk how badly the analogue of the section conjecture fails over finite fields.
Tevelev, Jenia
Geometric tropicalization and Cox coordinates
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I will try to explain how tropical geometry, Cox coordinates, and an equivariant flattening lemma can be used to study subvarieties of toric varieties. I will focus on the solution of the problem posed by B. Teissier: when is it possible to resolve singularities by a toric resolution of an ambient toric variety?
Voloch, Felipe
Finite descent obstructions for integral points on curves
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We will discuss various conjectures about finite descent obstructions for integral points on curves and their interrelations. We will also discuss an approach (via modularity) that leads to results on modular curves over the rational and can perhaps be extended to many other situations.

Participants

Name	Institution
Arzhantsev, Ivan	Lomonosov Moscow State University
Baeker, Hendrik	University of Tuebingen
Batyrev, Victor	Universität Tübingen
Borovoi, Mikhail	Tel Aviv University
Colliot-Thélène, Jean-Louis	CNRS et Université Paris-Sud
Demarche, Cyril	Université Paris 6
Derenthal, Ulrich	Ludwig-Maximilians-Universität München
Florence, Mathieu	Université Paris 6
Gayfullin, Sergey	Moscow State University
Gille, Philippe	Ecole Normale Supérieure
Harari, David	Universite Paris-Sud
Harpaz, Yonatan	The Hebrew University in Jerusalem
Hausen, Jürgen	Universität Tübingen
Herppich, Elaine	University of Tübingen
Kuyumzhiyan, Karine	Universite Grenoble I / Moscow State University
Laface, Antonio	Universidad de Concepción
Lee, Ting-Yu	DMA, ENS
Loughran, Daniel	University of Bristol
Mendes da Costa, David	University of Bristol
Mennecke, Susanne	Universität Tübingen
Pál, Ambrus	Imperial College London
Perepechko, Aleksandr	Institut Fourier
Pirutka, Alena	Universite Paris-Sud
Preu, Thomas	Zurich Graduate School in Mathematics
Serganova, Vera	UC Berkeley
Shlank, Tomer	Hebrew University In Jerusalem
Skorobogatov, Alexei N	Imperial College London
Stix, Jakob	Universität Heidelberg
Stoll, Michael	Universität Bayreuth
Testa, Damiano	University of Oxford
Tevelev, Jenia	University of Massachusetts at Amherst
Viray, Bianca	Brown University
Voloch, Felipe	University of Texas at Austin