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# SoSe 2012

Seminar: Algebraic Geometry Summer Semester 2012

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

Schedule:

 16.04.2014 No talk! 23.04.2012 Geoffrey Scott (Michigan) Sheaf Cohomology on T-Varieties Abstract: The definition of a T-varety is the same as that of a toric variety, except that the dimension of the torus may be less than the dimension of the variety. Motivated by the correspondences between the geometry of a toric variety and the combinatorics of its fan, we wish to translate properties of a T-variety into properties of the gadgets that classify them, called divisorial fans. In this talk, I will define these divisorial fans, review the way in which the cohomology of torus-invariant divisors on a toric variety can be calculated combinatorially, and show how this technique can be generalized to the T-variety case. 30.04.2012 Niels Lindner (HU Berlin) Plane cuspidal curves and Alexander polynomials Abstract: Given a curve in the complex projective plane, it is an interesting question to determine the fundamental group of its complement. An invariant of this group is given by the Alexander polynomial, which is closely related to the singularities of the curve. This relationship will be investigated in the case of cuspidal curves with the help of commutative algebra. 07.05.2012 Florian Geiß (Saarbrücken) Unirationality of Hurwitz Spaces and Ulrich Bundles Abstract: The unirationality of Hurwitz spaces of d-gonal curves of all genera g>=d-1 is classical for d<=5. We give a computer-aided proof of the unirationality of Hurwitz spaces of 6-gonal curves for most values of g<=45. We also present an application of this construction to the existence of Ulrich bundles, i.e. arithmetically Cohen-Macaulay bundles on a projective variety X that have the maximum number of generators of the associated graded module. We give a short overview of the mathematical context and outline the proof of existence of stable Ulrich bundles of any rank r>=2 on a general cubic threefold. The latter is based on work of M. Casanellas and R. Hartshorne and joined work with F.-O. Schreyer. 14.05.2012 Bernd Kreussler (Limerick, Ireland ) Fine analytic moduli spaces of sheaves on families of curves Abstract: This talk reports about joint work with Igor Burban. We consider a family of integral projective curves over a complex analytic space. We show that there exists a fine moduli space of relatively simple sheaves. This amounts to proving that the usual moduli functor is already a sheaf. To achieve this we prove a vanishing theorem for simple sheaves. 21.05.2012 Helge Ruddat (Mainz) Towards Mirror Symmetry for Varieties of General Type Abstract: Using Landau-Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau-Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main results (j.w. Gross, Katzarkov) shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers defined using a mixed Hodge structure on the hypercohomology of the perverse sheaf. This exchange can be anticipated from an analysis of Hochschild homology of the relevant categories arising in homological mirror symmetry in this case; we also conjecture that a similar, but different, exchange of dimensions arises from Hochschild cohomology, relating the cohomology of sheaves of polyvector fields on the hypersurface to the cohomology of the critical locus. 28.05.2012 Pfingsten! 04.06.2012 Robert Klinzmann (British Columbia, Canada) Grothendieck 2.0: derived algebraic geometry Abstract: Many moduli spaces, e. g., the moduli of vector bundles of surfaces, lack smoothness, a property that simplifies matters a lot. In the 90s Kontsevich claimed that singularities of moduli spaces arise due to truncations, e. g., of the cotangent complex, that we have implicitly performed during the construction of parameter spaces. We will illustrate this observation with the help of the moduli stack of vector bundles on a projective algebraic surface, present ideas that naturally give rise to the definition of Töen-Vezzosi \infty-stacks and give a first introduction into derived algebraic geometry and different realizations. 11.06.2012 Ihsen Yengui (Tunesia/Berlin) Algorithms for computing syzygies over polynomial rings over valuation rings 18.06.2012 Masoud Kamgar (Bonn) 2:15 pm !! Universal property of algebraic K-theory Abstract: Quilllen's higher algebraic K-theory is a fundamental tool with many applications in different branches of mathematics. Recently, using the language of higher category theory, the nature of K-theory has become better understood. In this talk, I will explain the universal property of K-theory of higher monoidal categories. Time permitting, I will also explain the cobordism hypothesis for classical field theories, using this universal property. 18.06.2012 Carl Tipler (Nantes, France) 4:15 pm !! Deformations of toric extremal manifolds Abstract: "Using the method of Székelyhidi, we reduce the problem of existence of extremal Kähler metrics on complex deformations of extremal Kähler manifolds to a finite dimensional GIT problem. We compute stable points in the case of toric manifolds, providing new examples of extremal Kähler surfaces. 25.06.2012 Lutz HIlle (Münster) Spherical Twists and Strict Tilting (joint with David Ploog) Abstract: The derived category of coherent sheaves on a rational surface is well understood in terms of full exceptional collections. We study a certain subcategory associated to a chain of (-2)-curves. This subcategory allows many autoequivalences, called spherical twists, and is isomorphic to a category of modules over the endomorphism algebra of k[T]/T^t. The principal aim of this talk is to classify all exceptional and spherical objects in this subcategory, and, moreover, also all tilting objects. We generalize this construction to other configurations. Eventually, we describe the relevant modules in terms of worm diagrams (see the talk on Thursday). This talk is closely related to my colloquium talk on Thursday, where I start with the elementary construction and explain more details explicitly. 02.07.2012 Milena Hering (Connecticut, USA) Algebraic properties of invariant rings Abstract I will review the first and second fundamental theorem for the action of the special linear group on the Grassmannian. I will then discuss these properties for the invariant ring describing the moduli space of n ordered points on the projective line and present some theorems about these invariant rings. This is joint work with Ben Howard. 09.07.2012 Dmitry A. Timashev (Moscow) Complexity and rank of Lagrangian subvarieties (joint work with V.S.Zhgoon) Abstract: Suppose that a connected reductive group G acts on a smooth complex algebraic variety X and Y is a G-stable smooth subvariety in X. An old result of Panyushev states that the normal and conormal bundles N and N^* of Y in X have the same complexity and rank as those of X. (Here the complexity of a G-variety is the codimension of a general B-orbit, where B is a Borel subgroup of G, and the rank is the rank of the lattice of eigenweights of all B-semi-invariant rational functions.) These two numerical invariants are of great importance in equivariant compactification theory, harmonic analysis, etc. We give a simple proof of Panyushev's theorem and provide some generalizations. To this end, we note that N^* is a G-stable Lagrangian subvariety in the cotangent bundle T^*X equipped with the canonical symplectic structure. First, we extend the theorem to general (G-stable) Lagrangian subvarieties in T^*X (which are quite close to conormal bundles). Secondly, we replace T^*X with an arbitrary symplectic algebraic variety M with a Hamiltonian action of G and consider a G-stable Lagrangian subvariety S in M. We express the complexity and rank of S in terms of certain invariants of the Hamiltonian action, which are used to replace the complexity and rank of X for M=T^*X. A prevailing idea of the proofs is to use deformation to normal bundles.