WS 2010/11

Seminar: Algebraic Geometry Winter Semester 2010/2011

Unless otherwise specified, all talks take place in room 119, Arnimallee 3 and begin at 16:30.

Schedule:

           
18.10.2010   Christine Berkesch (Purdue/Stockholm)
  Poset structures in Boij-Söderberg theory
  Abstract: Boij--Söderberg theory is the study of the cone of cohomology tables of coherent sheaves over projective spaces and the cone of standard graded minimal free resolutions over polynomial rings, which have simplicial fan structures induced by partial orders on their extremal rays. I will discuss a new interpretation of these partial orders in terms of the existence of nonzero homomorphisms. These results not only improve our understanding of the sheaves and modules corresponding to the extremal rays of these cones, but they also suggest the naturality of these partial orders and provide tools to study extensions of Boij--Söderberg theory to other projective varieties and nonstandard graded rings. This is joint work with Dan Erman, Manoj Kummini, and Steven V Sam.
   
25.10.2010  Stefan Guenther (FU-Berlin)
  Total differential calculus for algebraic varieties
   
01.11.2010 Vlad Lazic (Imperial College London)
  New outlook on Mori theory. 
  Abstract: I will give a gentle introduction into Mori theory and then sketch how finite generation of certain algebras (proved by a self-contained argument in a paper by Paolo Cascini and me) implies in a straightforward way all the main results of Mori theory. This gives a new and much more efficient organisation of the theory. This is joint work with Alessio Corti.
   
08.11.2010 Tom Coates (London)
  Mirror symmetry and Fano Varieties
  Abstract: A remarkable insight of Vasily Golyshev is that one should be able to use mirror symmetry to find and classify Fano varieties. I will explain the ideas involved and report on computer experiments that test Golshev's approach in dimension 3.
   
15.11.2010  No talk--North German Algebraic Geometry Seminar (Oldenburg, 18 - 19 November)
   
22.11.2010  Jan Christophersen (Oslo) 
  Deformations of monomial ideals
  Abstract: A report on work recently started with María Cruz Fernandez-Fernandez. We give a combinatorial description of first order deformations and obstructions. As possible application we wish to apply deformation theory to the multigraded Hilbert scheme of Haiman and Sturmfels.
   
29.11.2010  Ilarion Melnikov (AEI-MPG, Potsdam) 
  Deformations of gauged linear sigma models
  Abstract: Mirror symmetry, a relation between two topologically distinct Calabi-Yau manifolds, is an important tool in string theory.  To make use of this tool, we must have a good understanding of the map between the moduli spaces of the mirror manifolds.  In general it is difficult to construct this map explicitly; however, there are special situations where the map is under control.  An important example is provided by the Batyrev mirror pair construction.  Here, the "toric" subset of the Kaehler moduli is mapped to the "polynomial" subset of the complex structure moduli via the monomial-divisor mirror map.  The physical realization of this construction is via gauged linear sigma models.  In my talk I will describe some details of this construction, with particular emphasis on the mysteries of the "non-toric" and "non-polynomial" deformations.  I will also mention some natural bi-Hermitian geometries related to the recently introduced T-varieties.
   
06.12.2010 Anda Degeratu (AEI-MPG, Potsdam) 
  Invariants of elliptically fibered Calabi-Yau 3-folds
  Abstract: We look at a special type of elliptically fibered Calabi-Yau 3-folds arising via the heterotic--type II A string-string duality. We describe the imprint of this duality on the geometry and topology of the Calabi-Yau.
   
13.12.2010 David Ploog (Hannover)-- 14:15 - 15:45!
  Derived categories of toric surfaces
  Abstract:The derived categories of smooth, projective surfaces have been mainly investigated for their abundance of exceptional objects. However, they can also possess spherical objects. We show how the derived symmetries are standard apart from the associated spherical twists. This is used to present the spherical objects in terms of the exceptional ones. (Joint work with Nathan Broomhead.)
   
03.01.2011 Duco van Straten (Mainz) 
  Diagonals
  Abstract:The theory of diagonals offers an interesting approach to arithmetical propoerties of periods. In the talk we explain why the principle period of a Calabi-Yau 3-fold is an 8-diagonal.
   
10.01.2011 Elena Martinengo (FU-Berlin)--two talks: 14:00 - 18:30
  Deformations of complex manifolds via dgla.
  Abstract: I will start recalling basic tools of classical deformation theory and introducing the differential graded Lie algebras and L-infinity algebras approach to it. In particular, I will concentrate on the analysis of deformations of complex manifolds, making some classical examples. Then I will give a reinterpretation of Kodaira-Spencer approach by means of the language of stacks, I will review recent Getzler and Hinich?s results in this direction and present our refinement of them, obtained in a joint work with D. Fiorenza and M. Manetti. In the second part of the talk, I will start with the analysis of deformations of affine singular schemes in terms of dglas. As further development, I will briefly explain how the improvement of the tecniques presented can be applied to study deformations of a singular variety; this is part of a work in progress with D. Fiorenza and D. Iacono.
   
17.01.2011 Andreas Hoering (Paris/Freiburg)
  Anticanonical divisors and curve classes on Fano manifolds (joint work with Claire Voisin)
  Abstract: A classical theorem of Shokurov says that if X is a smooth Fano threefold and Y a general element in the anticanonical system, then Y is also smooth. In this talk I will explain a strategy to generalise this statement to Fanos of arbitrary dimension and prove an almost optimal result for fourfolds. I will also explain how to use this result to study certain Hodge classes on Fano fourfolds.
   
24.01.2011 Nathan Ilten (Bonn)
  Upgrades and Downgrades of p-divisors
  Abstract: K. Altmann and J. Hausen have shown that any affine T-variety can be described by a p-divisor, i.e. a divisor on some normal variety Y with polyhedral coefficients. In this talk, I will discuss how these p-divisors change when one "upgrades" or "downgrades" the torus action, i.e. when one enlarges the torus or restricts to a subtorus. I will also present an application of upgrades by showing how to recover a formula of Altmann and L. Petersen for the p-divisor of the Cox ring of a rational complexity-one T-variety. This is joint work with R. Vollmert.
   
31.01.2011 Stefan Kebekus (Freiburg)
  Differentailformen auf singulären Räumen
   
07.02.2011 Peter Scholze (Bonn)
  Perfectoid spaces
  Abstract: We will explain the theory of perfectoid spaces, which compares objects in characteristic p with objects in characteristic 0. For example, a toric variety over F_p((t)) can be regarded as a procover of the "same" toric variety over some infinite extension of Q_p, in some precise sense. This generalizes the theorem of Fontaine, that the absolute Galois groups of F_p((t)) and certain infinite extensions of Q_p are isomorphic, to higher dimensions.
   
14.02.2011 Lars Petersen (FU-Berlin)
  Okunkov bodies II