Seminar: Algebraic Geometry Winter Semester 2012/2013
Unless otherwise specified, all talks take place in room 119, Arnimallee 3 and begin at 16:00.
I will demonstrate how the computer programs Singular and Polymake interact when studying examples touching both discrete and algebraic geometry.
|Marcos Jardim (Campinas, Brazil)
|Hilbert scheme of points on affine spaces
|Abstract: The Hilbert scheme of points parametrizes 0-dimensional subschemes of the affine space. Nakajima gave a descrition of the 2-dimensional 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.
|Tobias Finis (FU-Berlin)
"Einführung in die Spurformel für reduktive Gruppen" (2:30 - 4:00)
" Eine kombinatorische Identität für Polyederfächer und ihre Anwendung auf die Superformel" (4:15 - 5:45)
|Konrad Schöbel (Jena)
|Separationskoordinaten und Modulräume stabiler Kurven
|Abstract: Separation coordinates are coordinates in which the classical Hamilton-Jacobi 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 non-trivial example: the 3-sphere. In particular, we show that the moduli space of separation coordinates on the 3-sphere is naturally isomorphic to a certain moduli space of stable curves with marked points. Several generalisations of this result will be proposed.
|Alwaleed Kamel (FU-Berlin)
|S_4-Geometry of 2-Weierstrass points on Kuribayashi quartic curves
|Abstract: We study the geometry of the 2-Weierstrass points on the Kuribayashi quartic curves: Ca :x4+y4+z4+a(x²y²+y²z²+x²z²)=0 (a≠1,±2). The 2-Weierstrass points on Ca are divided into flexes and sextactic points. It is known that the symmetric group S4 acts on Ca (See ). Using the S4-action, we classify the 2-Weierstrass points on Ca.
References:  Kuribayashi,A. and Sekita,E.:On a family of Riemann surfaces I, Bull. Fac. Sci. Eng. Chuo Univ. 22 (1979), 107-129.
|Jaroslaw Wisniewski (Warschau)
|Coordinate rings of resolutions of quotient singularities
|Abstract: I will report on ongoing project with Marysia Donten-Bury which aims at understanding quotient symplectic 4-dimensional singularities.
|Kaie Kubjas (FU-Berlin)
|Toric Degenerations of Conformal Block Algebras
|Abstract: Jukes-Cantor 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.
|Conformal block algebras, Berenstein-Zelevinsky triangles and group-based 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.
|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 so-called 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).
|Jürgen Hausen (Tübingen)
|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.
|David Ploog (Hannover)
|14:30 - 17:45!
|Singularity categories (Buchweitz and Orlov)
|Abstract: Buchweitz defined in 1986 the stable category of a Cohen-Macaulay ring, a triangulated category which generalises Eisenbud's concept of matrix factorisations. In 2004, Orlov re-invented 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.
|No talk! (double talk on January 7th)
|No talk! (double talk on January 28th)
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 so-called colored fans and, finally, we show how the Tits fibration can be used to understand spherical varieties as T-varieties. Thus, colored fans turn into p-divisors. 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.
|Klaus Altmann: (FU Berlin)
|T-actions 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 so-called colored fans and, finally, we show how the Tits fibration can be used to understand spherical varieties as T-varieties. Thus, colored fans turn into p-divisors. The latter is joint work with Valentina Kiritchenko and Laars Petersen.