SoSe 2014

Seminar: Algebraic Geometry

Unless otherwise specified, all talks take place in room 119, Arnimallee 3 and begin at 16:15.

Schedule:       

14.04.2014  No talk!
   
21.04.2014 No talk--Easter Monday
   

28.04.2014

&

05.05.2014

Lars Kastner (FU-Berlin)
  Ext-Algebras on cyclic quotient singularities
  Abstract: Cyclic quotient singularities are toric varieties arising from two-dimensional cones. Given two T-invariant Weil divisors D and D' on a cyclic quotient singularity, we want to study the modules Ext^i(D,D'). The T-action is reflected by a multigrading on these modules which allows derivation of a combinatorial description. Yoneda's description of these modules as equivalence classes of exact sequences equips the direct sum of all Ext^i(D,D) for increasing i with the structure of an algebra. Thus we want to extend the combinatorial description of the single summands to gain an understanding of the algebra structure as well.
   
12.05.2014 Nero Budur (Leuven)
  Cohomology jump loci
  Abstract:In practice, any deformation problem over fields of characteristic zero is governed by a differential graded Lie algebra (DGLA). Following Deligne, Goldman-Millson and Simpson described the local structure of deformation spaces for various geometric situations. Given an object with a notion of cohomology theory, how can one describe all its deformations subject to cohomology constraints? We will present an approach via DGLA pairs. As applications we discuss the structure of cohomology jump loci of vector bundles and of local systems. This is joint work with Botong Wang.
   
19.05.2014 Alex Constantinescu (FU)
  Gröbner cells in the Hilbert scheme of points  
  Abstract: We present a parametrization of some Gröbner cells in the Hilbert scheme of points in the affine plane. This parametrization extends to a dense subset of the Hilbert scheme of points in the projective plane. We then use this parametrization to describe the Betti strata of this Hilbert scheme, and discuss possible applications to the study extension algebras.
   
26.05.2014 Mateusz Michalek (Warschau/FU)
  Matroids and toric geometry 
  Abstract: Matroids are combinatorial structures that generalize various notions of independence: e.g. linear or algebraic. We will start by presenting a few equivalent definitions of matroids. To any matroid one naturally associates a normal, projective toric variety. Many combinatorial properties of matroids can be expressed in terms of the associated toric varieties. One of the examples is White's conjecture, which predicts generators of the ideal of the associated toric variety. We will prove that these generators indeed define the correct projective scheme. We will also show White's conjecture for strongly base orderable matroids. As an application, we provide a scheme-theoretic description of the closure of any torus orbit on any Grassmannian. These results are from a joint work with Michal Lason.
   
02.06.2014 Imran Qureshi (Lahore / FU-Berlin)
  Polarized 3-folds in weighted flag varieties
  Abstract: Let G be a reductive Lie group and P be a Parabolic subgroup of G, then the quotient $ \Sigma= G/P$ is a called a flag variety, a projective subvariety of the projectivization of some irreducible G-representation V . I will describe the notion of the weighted flag variety by using combinatorial root data of G. Then we show that how one may usethese varieties as an ambient spaces to construct interesting classes of polrized 3-folds, such as Calabi-Yau 3-folds and Fano 3-folds with mild singularities. Time permitting, we present an algorithm to compute all possible 3-folds of fixed class with isolated singularities in a fixed weighted flag variety.
   
09.06.2014 No talk--Pentecost!
   
16.06.2014 Vikraman Balaji (Chennai Mathematical Institute, India)
 at 14:!5!! Degenerations of moduli space of Hitchin pairs
  Abstract:  In this talk I will discuss some recent results of mine with Nagaraj and Barik on Degenerations of the moduli space of Hitchin pairs.This leads to some new compactifications of Picard varieties of stablecurves different from the ones studied by Oda-Seshadri or Caporaso.
   
23.06.2014 Anthony Iarrobino (NEU, Boston)
  When do two nilpotent matrices commute?
 

Abstract:The similarity class of an n X n nilpotent matrix B over a field k is given by its Jordan type, the partition P of n, specifying the sizes of the Jordan blocks. The variety N(B) parametrizing nilpotent matrices that commute with B is irreducible, so there is a partition Q= Q(P) that is the maximum commuting orbit of P: that is, Q(P) is the generic Jordan type for matrices A in N(B). Q(P) has parts that differ pairwise by at least two and is ``stable'': Q(Q(P))=Q(P).

We review what is known about the map P to Q(P), in particular a digraph determined by P, and a recursive conjecture by P. Oblak (2008), very recently shown by R. Basili after partial results by P. Oblak, T. Kosir, L. Khatami, D.I. Panyushev, and others.

We then discuss a ``Table theorem" when Q has two parts and a ``Box Conjecture" in general for the set of partitions P having a given partition Q as maximum commuting orbit.

This work is joint with Leila Khatami, Bart van Steirteghem, and Rui Zhao.

   
30.06.2014 Jan Christophersen (Oslo)
  Local cohomology and higher cotangent cohomology for cones over Grassmannians 
  Abstract: This is joint work with Nathan Ilten. Let A be the homogeneous coordinate ring of a Grassmannian Grass(r,n) in the Plücker embedding. We use representation theory to compute higher cotangent cohomology for A. If d = r(n-r)+1 = dim A then the cotangent cohomology vanishes in degree 1 to d. The degree d part vanishes if and only if r=2 or n-2. This has applications for families of complete intersections in Grassmannians. Possible generalizations to isotropic Grassmannians and other homogeneous spaces will be discussed.
   
07.07.2014 Alexander M. Kasprzyk (London)
  Mirror symmetry and mutations of lattice polytopes  
  Abstract In this talk I aim to summarise the recent joint work of Coates, Corti, Galkin, Golyshev, and myself, attempting to classify Fano manifolds via mirror symmetry. In particular I will describe the process of mutation, a compelling combinatorial operation on lattice polytopes that has immediate applications to the study and classification of polygons.
   
14.07.2014  Volkmar Welker (Marburg)--talk cancelled!