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 wellknown for special ground fields to arbitrary nonarchimedean 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 pdivisors  
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, algebraischgeometrische 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 metamathematical principle: Objects that saturate a natural inequality have distinguished discrete, arithmetic and combinatorial nature. First example is MasonStothers inequality and Belyi functions. Second example is Szpiro(BeauvilleMirandaPerssonShiodaTate) inequality and extremal elliptic surfaces. Third example is a natural linearizationgeneralization 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 groupbased 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 CalabiYau 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 (FUBerlin) 
The Cone of Hilbert Functions in the nonstandard graded Case  
Abstract: Motivated by the BoijSöderberg 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 rth secant variety to the duple 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 settheoretically if and only if the Hilbert scheme parametrising 0dimensional 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 Amodules grmodA. On the one hand, there is the derived category of the quotient of grmodA by torsion, which gives the derived category of Proj(A) (by Serre in the commutative and by ArtinZhang in the noncommuative setting). But you can also consider the derived category of grmodA modulo the derived category generated by the free modules, which is the socalled 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 noncommutative projective schemes.  