Seminar: Algebraic Geometry Winter Semester 2012/2013
Unless otherwise specified, all talks take place in room 119, Arnimallee 3 and begin at 16:00.
Schedule:
I will demonstrate how the computer programs Singular and Polymake interact when studying examples touching both discrete and algebraic geometry.
15.10.2012  Marcos Jardim (Campinas, Brazil) 
Hilbert scheme of points on affine spaces  
Abstract: The Hilbert scheme of points parametrizes 0dimensional subschemes of the affine space. Nakajima gave a descrition of the 2dimensional case in terms of linear algebraic data and monads. Our goal is to generalize Nakajima's description for higher dimensional affine spaces. Joint work with Patricia B dos Santos e Amar Henni.  
22.10.2012  Tobias Finis (FUBerlin) 
"Einführung in die Spurformel für reduktive Gruppen" (2:30  4:00) and " Eine kombinatorische Identität für Polyederfächer und ihre Anwendung auf die Superformel" (4:15  5:45) 

29.10.2012  No talk!! 
05.11.2012  Konrad Schöbel (Jena) 
Separationskoordinaten und Modulräume stabiler Kurven  
Abstract: Separation coordinates are coordinates in which the classical HamiltonJacobi equation can be solved by a separation of variables. We establish a new and purely algebraic approach to the classification of separation coordinates under isometries. This will be made explicit for the least nontrivial example: the 3sphere. In particular, we show that the moduli space of separation coordinates on the 3sphere is naturally isomorphic to a certain moduli space of stable curves with marked points. Several generalisations of this result will be proposed.  
12.11.2012  Alwaleed Kamel (FUBerlin) 
S_4Geometry of 2Weierstrass points on Kuribayashi quartic curves  
Abstract: We study the geometry of the 2Weierstrass points on the Kuribayashi quartic curves: C_{a} :x^{4}+y^{4}+z^{4}+a(x²y²+y²z²+x²z²)=0 (a≠1,±2). The 2Weierstrass points on C_{a} are divided into flexes and sextactic points. It is known that the symmetric group S_{4} acts on C_{a} (See [1]). Using the S_{4}action, we classify the 2Weierstrass points on C_{a}. References: [1] Kuribayashi,A. and Sekita,E.:On a family of Riemann surfaces I, Bull. Fac. Sci. Eng. Chuo Univ. 22 (1979), 107129. 

19.11.2012  Jaroslaw Wisniewski (Warschau) 
Coordinate rings of resolutions of quotient singularities  
Abstract: I will report on ongoing project with Marysia DontenBury which aims at understanding quotient symplectic 4dimensional singularities.  
26.11.2012  Kaie Kubjas (FUBerlin) 
Toric Degenerations of Conformal Block Algebras  
Abstract: JukesCantor binary model is a statistical model that associates with a tree a toric variety. Buczynska and Wisniewski showed that any two such varieties associated with trivalent trees with the same number of leaves are deformation equivalent, that is, they lie on the same connected component of the Hilbert scheme of the projective space. However, the question was left open whether all toric varieties associated with trees with the same number of leaves lie on the same irreducible component of the Hilbert scheme, and what is the general point on that component. Sturmfels and Xu answered this question by constructing sagbi deformations of the toric varieties. Manon has extended their result in several different directions using conformal block algebras. In this talk, I will review these results and introduce some open questions.  
03.12.2012  Kaie Kubjas 
Conformal block algebras, BerensteinZelevinsky triangles and groupbased models  
Abstract: This talk will be independent of the previous talk. No previous knowledge about conformal block algebras is expected. I will concentrate on combinatorial and representation theoretical aspects of conformal block algebras.  
10.12.2012  Jasmin Matz (Bonn) 
An explicit bound for global coefficients in Arthur's trace formula for GL(n)  
Abstract: Arthur's trace formula is an important tool in number theory and harmonic analysis and describes a connection between geometric properties of a reductive group and its representations. After giving some necessary background, we will discuss an explicit upper bound for the socalled global coefficients appearing in the trace formula in the case of GL(n). These coefficients are in general left unspecified, but a better understanding of them is essential for applications. At the end, we shall also discuss an anticipated application to the Weyl's law for Hecke operators on GL(n).  
13.12.2012  Jürgen Hausen (Tübingen) 
Donnerstag!!  Complete intersection Fano varieties andpolyhedral complexes 
Abstract: Generalizing the correspondence between toric Fano varieties and lattice polytopes, we associate to any Fano variety with a complete intersection Cox ring its ``anticanonical complex'', which is a certain polyhedral complex living in the lattice of an ambient toric variety. For resolutions constructed via the tropical variety, the lattice points inside the anticanonical complex control the discrepancies. The construction applies in particular to Fano varieties with a torus action of complexity one and there it leads, for example, to simple characterizations of terminality and canonicity.  
07.01.2013  David Ploog (Hannover) 
14:30  17:45!  Singularity categories (Buchweitz and Orlov) 
Abstract: Buchweitz defined in 1986 the stable category of a CohenMacaulay ring, a triangulated category which generalises Eisenbud's concept of matrix factorisations. In 2004, Orlov reinvented this in a geometrical setting and defined the singularity category of a variety. In my talks, I will motivate and introduce these notions. I will give examples and treat Knoerrer periodicity. Die Form des Doppelvortrages enstand aus der Idee, dass in diesem Vortrag die erlaubten Techniken und fallstricke der triangulierten Kategorien genau erlaeutert werden sollen. 

14.01.2013  No talk! (double talk on January 7th) 
21.01.2013  No talk! (double talk on January 28th) 
28.01.2013
16:15  17:45 
Ana Maria Botero (FU Berlin) 
Spherical varieties  an introduction  
Abstract: First, we will introduce the notion of spherical varieties and discuss important subclasses (horospherical, toroidal, sober, symmetric) and many examples of them. We present their description by socalled colored fans and, finally, we show how the Tits fibration can be used to understand spherical varieties as Tvarieties. Thus, colored fans turn into pdivisors. The latter is joint work with Valentina Kiritchenko and Lars Petersen.  
18:15  19:15  Lars Kastner (FU Berlin) 
Polymake and Singular  
Abstract: I will demonstrate how the computer programs Singular and Polymake interact when studying examples touching both discrete and algebraic geometry.  
04.02.2013  Klaus Altmann: (FU Berlin) 
Tactions on spherical varieties  
Abstract: First, we will introduce the notion of spherical varieties and discuss important subclasses (horospherical, toroidal, sober, symmetric) and many examples of them. We present their description by socalled colored fans and, finally, we show how the Tits fibration can be used to understand spherical varieties as Tvarieties. Thus, colored fans turn into pdivisors. The latter is joint work with Valentina Kiritchenko and Laars Petersen.  
11.02.2013  No talk! 