Oberseminar Algebraische Geometrie (SS 2012)

Ivan Arzhantsev - Werner Bley - Ulrich Derenthal - Andreas Rosenschon - Eva Viehmann

Summer 2012, Wednesdays, 16:30, room B 040 (Mathematisches Institut, Theresienstr. 39, 80333 München)

Previous semester: Winter 2011/12 - next semester: Winter 2012/13

Abstracts

Fei Xu (Capital Normal University, Beijing): Very strong approximation with Brauer-Manin obstruction for certain algebraic varieties (18.4.12)

By using Manin's idea, one can refine the classical strong approximation to the strong approximation with Brauer-Manin obstruction. Several achievements have been made recently for homogeneous spaces and some families of homogeneous spaces with some application to study the existence of the integral points. In this talk, we will explain that for certain algebraic varieties one can have even stronger property than strong approximation with Brauer-Manin obstruction. We will also discuss a conjecture proposed by Harari and Voloch.

Ivan Arzhantsev (Moscow State University / LMU München): Additive structures on projective varieties and local algebras (25.4.12)

Let C be the additive group of the field of complex numbers. An additive structure on an n-dimensional complex projective variety X is a regular action of Cⁿ on X with an open orbit. Such a structure allows to consider X as an equivariant compactification of Cⁿ. This way we obtain an additive version of toric geometry.

In 1999 Hassett and Tschinkel established a remarkable correspondence between additive structures on projective spaces and Artinian local algebras. This correspondence implies that the number of equivalence classes of additive structures on Pⁿ with n>5 is infinite.

We will discuss the Hassett-Tschinkel correspondence and related numerical invariants of local algebras together with some classification results. A generalization of the correspondence to projective hypersurfaces will be given and additive structures on flag varieties of semisimple algebraic groups will be described. Also we will discuss some results on equivariant compactifications of arbitrary commutative linear algebraic groups.

Kazim Buyukboduk (Koç University): Deformations of Kolyvagin systems (9.5.12)

Mazur's theory of Galois deformations, inspired by Hida's earlier work on families of modular forms, has led to the resolution of many important conjectures in Number Theory, such the Taniyama-Shimura conjecture or many cases of the strong Artin conjecture for GL₂. In this talk, I will first give a general outline of Mazur's abstract theory and talk about a recent result that Kolyvagin systems (which Mazur and Rubin prove to exist for mod p Galois representations) do often deform to a big Kolyvagin system for the "Universal Galois deformation" representation. I will touch upon arithmetic applications of this result.

Ulrich Görtz (Universität Duisburg-Essen): Affine Deligne-Lusztig varieties in the Iwahori case (16.5.12)

Affine Deligne-Lusztig varieties are a group-theoretic tool to study arithmetic properties of Shimura varieties. They are analogs in the setting of an affine root system of classical Deligne-Lusztig varieties, which have significant applications in the representation theory of finite groups of Lie type.

I will explain the basic notions, some results and open questions in elementary terms, and report on recent work with Xuhua He and Sian Nie.

Marc Nieper-Wißkirchen (Universität Augsburg): Smith Theory and Irreducible Holomorphic Symplectic Manifolds (13.6.12)

Smith theory is the study of the cohomological properties of a G-space X, its invariant subspace X^G, and its quotient space X/G where G is a finite group of prime order. We derive a precise relation between the mod-p cohomology of X^G and natural invariants of the integral cohomology of X. We apply these results to irreducible holomorphic symplectic manifolds whose deformation type is the Hilbert scheme of two points on a K3 surface.

The talk will naturally include a mixture of topological and algebro-geometrical (in characteristic 0) methods. We will explain the most important concepts (e.g., equivariant cohomology, group cohomology, irreducible holomorphic symplectic manifolds).

Alfred Weiss (University of Alberta): On equivariant Iwasawa theory (20.6.12)

This theme generalizes and refines the Main Conjecture of classical Iwasawa theory. The goal of the talk is to introduce its 'main conjecture' and to survey the main tools, especially the newer ones, of the 3 stages of its proof (when mu is 0, as conjectured by Iwasawa). This is joint work with J. Ritter.

Mathias Lederer (Universität Bielefeld): The Connect4 morphism for Hilbert schemes of points (27.6.12)

The Hilbert scheme of n points is the moduli space of ideals I in S = k[x₁, ..., x_d] of codimension n. The Gröbner deformation of such an I is a monomial ideal M_Δ in S. We will study H^Δ, the the moduli space of ideals I whose lexicographic Gröbner deformation is M_Δ. We will count the components of an open part of H^Δ, and determine its dimension, thus proving a conjecture of Sturmfels. The crucial notion is Connect4, a combinatorial operation on monomial ideals which enables us to distinguish components in the geometric object H^Δ. We will discuss a conjectural connection to the ring of symmetric functions.

Chad Schoen (Duke University): Curves on Threefolds (4.7.12)

This will be a (partial) survey of the area in algebraic geometry suggested by the title. In the selection of particular topics the emphasis will be on keeping prerequisites minimal and on connecting to classical material.

Urs Hartl (Universität Münster): The Hodge conjecture over function fields (18.7.12)

Over a global function field one has a neutral Tannakian category of mixed motives, namely the uniformizable mixed t-motives defined by G. Anderson. R. Pink clarified the concept of Hodge structures in equal characteristic and defined Hodge realizations of t-motives, by using the theory of σ-bundles on the rigid analytic punctured open unit disc. We prove the analog of the famous Hodge conjecture in this situation, namely that the Hodge realization functor induces an isomorphism of the motivic Galois group of a t-motive onto its Hodge group.

Yuri Prokhorov (Moscow State University): Subgroups of Cremona groups (18.7.12)

The Cremona group Cr_n is the group of automorphisms of the rational function field k(t₁,t₂,..., t_n). I describe the method that allows to find finite subgroups of Cr_n. I introduce minimal models (G-Fano-Mori models), outline the classification in dimension 2, and give a lot of examples in dimension 3. Relation to the notion of essential dimension will also be discussed.