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# WS 2013/2014

Seminar: Algebraic Geometry Winter Semester 2013/2014

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

Schedule:

 14.10.2013 Georg Hein (Essen) Orthogonalität und Birationalität von Modulräumen 21.10.2013 Maria Donten-Bury (Warsaw) Cox rings of minimal resolution of surface quotient singularities Abstract: We study Cox rings of minimal resolutions of quotients of the plane by an action of a finite subgroup of GL(2,C). We will show two methods of describing these rings. The first one is to give the equation for the spectrum, which is a hypersurface in an affine space. The second is to give a set of generators of the Cox ring in a simpler ring associated with the investigated singularity. 28.10.2013 Martin Kalck (Bielefeld)  TALK CANCELLED! Cluster categories Abstract:  We give an introduction to cluster categories and explain how Auslander's algebraic McKay correspondence may be interpreted in this language. 04.11.2013 Rui Sun (Berlin/Munich) 14:15!!! T-duality in string theory and double field theory Abstract: I will start from the basis of T-duality in string theory, introduce how T-duality connects the different string theories and map them to M-theory. In the end I will talk about the T-duality in double field theory, especially how T-duality connects the geometric and non-geometric fluxes in double field theory. By introducing T-duality, I will try to give a general picture of string theory and double field theory which is also the topic of my doctoral project. 11.11.2013 Björn Andreas (FU-Berlin) Existence of stable sheaves on Calabi-Yau threefolds Abstract: We discuss the existence problem of stable sheaves on Calabi-Yau threefolds with prescribed Chern classes and a related conjecture by Douglas, Reinbacher and Yau. Evidence for the truth of the conjecture will be given. 18.11.2013 Jan Christophersen (Oslo) TALK CANCELLED! Comparing deformations via reductive group quotients Abstract: I will  compare  deformations of algebras to deformations of schemes in the setting of invariant theory. This is joint work with Jan Kleppe. We study deformations (abstract and embedded) of a scheme X which is a good quotient of a quasi affine scheme in some Spec S by a linearly reductive group G andcompare them to invariant deformations of S. This generalizes known results for comparing deformations of Proj S with those of S. As a bi-product we get results on generalized Euler sequences on X via the group action. 25.11.2013 June Huh (Michigan / MPI Bonn) Rota's conjecture, and positivity of algebraic cycles in the permutohedral toric variety Abstract: Rota's conjecture predicts that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. I will outline a proof for representable matroids using the Bergman fan. The same approach to the conjecture in the general case (for possibly non-realizable matroids) leads to several intriguing questions on higher codimension algebraic cycles in the toric variety associated to the permutohedron. 02.12.2013 Winfried Bruns (Osnabrück) 16:15 On the computation of generalized Ehrhart series and integrals in Normaliz Abstract: We describe an algorithm for the computation of generalized (or weighted) Ehrhart series based on Stanley decompositions as implemented in the offspring NmzIntegrate of Normaliz. The algorithmic approach includes elementary proofs of the basic results. We illustrate the computations by examples from combinatorial voting theory. 02.12.2013 Dragos Deliu (Wien) 18:00 Homological Projective Duality via Variation of VGIT Quotients Abstract: I will explain how to construct a homological projective dual to a GIT quotient provided we are also given the data of an elementary wall crossing. I will then show how this construction allows one to obtain a homological projective dual variety to the Veronese embedding of degree d. 09.12.2013 Bernd Sturmfels (Berkeley / MPI Bonn) Tropicalization of Classical Moduli Spaces Abstract: Algebraic geometry is the study of solutions sets to polynomial equations. Solutions that depend on an infinitesimal parameter are studied combinatorially by tropical geometry. Tropicalization works especially well for varieties that are parametrized by monomials in linear forms. Many classical moduli spaces (for curves of low genus and few points in the plane) admit such a representation, and we here explore their tropical geometry. Examples to be discussed include the Segre cubic, the Igusa quartic, the Burkhardt quartic, and moduli spaces of marked del Pezzo surfaces. Matroids, hyperplane arrangements, and Weyl groups play a prominent role. Our favorites are E6, E7 and G32.  This is joint work with Qingchun Ren and Steven Sam. 16.12.2013 Björn Andreas (FU-Berlin) On Anomalies and Euler-characteristics Abstract: We review the relation between physical anomalies and Euler-characteristics of Calabi-Yau manifolds and point to various open computational problems in this context. 06.01.2014 Andras Nemethi (Budapest) The geometric genus of hypersurface singularities Abstract:  Using the path lattice cohomology we provide a conceptual topological characterization of thegeometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non--degenerate hypersurfacesingularities. The first part of the talk will be a historical overview of the guiding problem regarding such characterizations. 13.01.2014 David Ploog (Essen) Discrete derived categories Abstract: Finite-dimensional algebras with discrete derived categories were introduced by Vossieck and studied by Bobinksi, Geiss, Skrowonksi. They are not hereditary but "almost as good". I will say what they are and what we can prove about them. Our goal is to study the stability manifold of thesecategories; we can prove that it is contractible. (Joint work with Nathan Broomhead and David Pauksztello.) 20.01.2014 Hans-Christian Graf von Bothmer (Hamburg) Rationality of algebraic hypersurfaces Abstract: In linear algebra we often solve a linear equation $\sum_{i=1}^n a_ix_i$ by giving a parametrization, for example $x_n = -\sum_{i=1}^{n-1} \frac{a_i}{a_n}x_i$ if $a_n \not=0$.Similarly one would like to solve algebraic equations, for example $x^3+y^3+z^3+xyz = 0$ by giving a parametrization of the form $z = polynomial(x,y)$. Unfortunately this is often impossible.A more modest question would be to ask wether a parametrization is possible for a given equation and if so find one. Even this question is a surprisingly difficult problem leading to a lot of nice geometry. In this talk I will present classical and modern results as well as open questions on this topic. 27.01.2014 Priska Jahnke (Darmstadt) Non-free lines on homogeneous complete intersections Abstract: In linear algebra we often solve a linear equation $\sum_{i=1}^n a_ix_i$ by giving a parametrization, for example $x_n = -\sum_{i=1}^{n-1} \frac{a_i}{a_n}x_i$ if $a_n \not=0$. Similarly one would like to solve algebraic equations, for example $x^3+y^3+z^3+xyz = 0$ by giving a parametrization of the form $z = polynomial(x,y)$. Unfortunately this is often impossible. A more modest question would be to ask wether a parametrization is possible for a given equation and if so find one. Even this question is a surprisingly difficult problem leading to a lot of nice geometry. In this talk I will present classical and modern results as well asopen questions on this topic. 03.02.2014 No talk! 10.02.2014 Anand Sawant (Tata Institute, Mumbai) Connectedness in A^1-homotopy theory of schemes Abstract:  A^1- (or motivic) homotopy theory is a homotopy theory for schemes developed by Morel and Voevodsky in 1990's in which the affine line plays the role of the unit interval.  In this lecture, we will study the sheaf of A^1-connected components of a scheme.  There are various notions of connectedness in this theory and it is natural to ask whether these objects have nice properties such as homotopy invariance and whether they are birational invariants of smooth proper schemes.  We will discuss conjectures of Morel and Asok-Morel regarding A^1-connected components of a scheme.  This is a report on a joint work with Chetan Balwe and Amit Hogadi.