WS 2011/12

Seminar: Algebraic Geometry Winter Semester 2011/12

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

Schedule:

           
17.10.2011 Walter Gubler (Regensburg)
  A guide to tropicalizations over valued fields
  Abstract: Tropicalizations form a bridge between algebraic and convex geometry. In this talk, we generalize basic results from tropical geometry which are well-known for special ground fields to arbitrary non-archimedean valued fields.
   
24.10.2011 Andre Mialebama Boesso (FU, Dakar)
  Noncommutative Gröbner basis
  Abstract: In this talk we will construct a noncommutative Gröbner basis in polynomial free associative algebra over a noetherian Valuation ring and solve the ideal membership problem
   
31.10.2011 Lutz Hille (Münster)
  Tilting Bundles on Rational Surfaces and Resolutions of the Diagonal
  Abstract: We start with the proof of the existence of a tilting bundle on a rational surface and describe its endomorphism algebra. Such a tilting bundle can be obtained as a universal extension of a sequence of line bundles, we give an introduction to universal extensions and present several applications. In the last part we discuss the exterior powers of the cotangent sheaf, there exist canonical complexes constructed from the tilting bundle representing the cotangent sheaf. Finally we discuss further applications for spherical twists and curves of negative selfintersection.
   
7.11.2011 Mats Boij (Stockholm)
  Cones of invariants and parameter spaces of modules
  Abstract:When studying Betti tables of modules, it turned out to be fruitful to study them not only as arrays of integers, but to look at the rational cone that is spanned by these arrays. This can also be done for Hilbert functions, and it might come as a surprise that there is anything new to say about them since the possible Hilbert functions of standard graded algebras were characterized by Macaulay in 1927. Part of this is joint work with Gregory G. Smith. I will also explain how the knowledge that came from the characterization of Betti tables up to scaling can be used in the study of parameter spaces of graded modules with a fixed Hilbert function.
   
14.11.2011 Yuri Tschinkel (NYU)
  Almost abelian anabelian geometry
  Abstract:I will explain some constructions from projective geometry which play an important role in the reconstruction of function fields of algebraic varieties from their Galois groups (joint work with F. Bogomolov).
   
21.11.2011 Robert Vollmert (FU)
  Deformations of toric varieties as p-divisors
  Abstract: I will show how certain deformations of toric varieties arise naturally as deformations of polyhedral divisors.
   
28.11.2011 Anton Deitmar (Tübingen)
  Gibt es einen Körper mit einem Element?
  Abstract: Um Methoden der algebraischen Geometrie auf zahlentheoretische Fragestellungen anwenden zu können, möchte man Spec Z als eine "Kurve" betrachten. Diese müsste dann über einem Körper der "Charakteristik Eins" definiert sein, den es nicht gibt. In den vergangenen zehn Jahren haben sich verschiedene Autoren bemüht, algebraisch-geometrische Konstrukte zu finden, die die Rolle dieses "Körpers mit einem Element" übernehmen könnten. In dem Vortrag wird über die verschiedenen Ansätze und bisherige Ergebnisse berichtet.
   
05.12.2011 Sergie Galkin (Tokyo)
  Inequalities and their extremes
  Abstract: I'll demonstrate a few interrelated (some famous and some new) incarnations  of the following meta-mathematical principle: Objects that saturate a natural inequality have distinguished discrete, arithmetic and combinatorial nature. First example is Mason-Stothers inequality and Belyi functions. Second example is Szpiro(-Beauville-Miranda-Persson-Shioda-Tate) inequality and extremal elliptic surfaces. Third example is a natural linearization-generalization of the second one:  Golyshev's inequality and extremal local systems. If time permits I'll show some other incarnations or explain why mirror reflections of Fano varieties is a generalizations of the inequality of arithmetic and geometric means.
   
12.12.2011 Mateusz Michalek (Grenoble)
  Toric varieties and Markov processes on trees
  Abstract: We will present a construction and properties of special toric varieties arising from Markov processes on trees. We will focus on so called group-based models. In this setting the interplay between combinatorics and toric geometry is even stronger than usually. A few open problems will be presented > as well as recent advancements. The approach that we will use should interest toric and in general algebraic geometers.
   
02.01.2012 No talk!
   
09.01.2012 No talk!
   
16.01.2012 Balazs Szendröi (Oxford)
  Cohomology of moduli of sheaves on Calabi-Yau threefolds
   
23.01.2012 Tarig Abdegadir (Bonn)
  Refined quiver representations of the McKay quiver
  Abstract: Take G an abelian subgroup of SL(n,C) (n<=3). Craw and Ishii show varying the stability condition on a moduli space of McKay quiver representations recovers all projective crepent resolutions of the singularity A^n/G. In this talk we will recover the stack [A^n/G] (a noncommutative crepant resolution) from the McKay quiver. We will then relate it to a commutative resolution via a change of stability conditions on a moduli space of refined representations of the McKay quiver.
   
30.01.2012 Marianne Merz (FU-Berlin)
  The Cone of Hilbert Functions in the non-standard graded Case
  Abstract: Motivated by the Boij-Söderberg theory we study the cone of the Hilbert functions of artinian modules finitely generated in degree 0 over the polynomial ring R=k[x,y] with the grading deg(x)=1 and deg(y)=n. We can describe the cone by the extremal rays and give a decomposition algorithm.
   
06.02.2012  Jarek Buczynski (Warsaw)
  Defining equations of secant varieties to Veronese reembeddings
  Abstract: We fix a projective variety X \subset P^n and an integer r. We are interested in the defining equations of the r-th secant variety to the d-uple Veronese reembedding of X, and we assume d is sufficiently large. (One of the interesting cases is, when X =P^n). With these assumptions we prove that the (r+1)-minors of a single matrix with linear entries are sufficient to define the secant variety set-theoretically if and only if the Hilbert scheme parametrising 0-dimensional Gorenstein subschemes of X of degree r is irreducible. In particular, if X is smooth and either dim X is at most 3 or r is at most 10, then the minors are sufficient. If dim X is at least 4 and r is sufficiently large, then the locus defined by the minors has some extra components.
   
13.02.2012  Andreas Hochenegger (Hannover)
 

Orlov's Theorem

  Abstract: Given a graded algebra A there are two interesting triangulated categories associated to the category of graded A-modules grmod-A. On the one hand, there is the derived category of the quotient of grmod-A by torsion, which gives the derived category of Proj(A) (by Serre in the commutative and by Artin-Zhang in the non-commuative setting). But you can also consider the derived category of grmod-A modulo the derived category generated by the free modules, which is the so-called graded triangulated category of singularities. Orlov showed that these two categories are quite closely connected. I will give a streamlined version of the original proof without using the machinery of non-commutative projective schemes.