# WS 2014/15

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

# Schedule:

**20 October 2014: Simon Keicher (Tübingen)**

**Algorithms for Mori dream spaces**

**Abstract: **Hu/Keel defined Mori dream spaces as algebraic varieties with finitely generated Cox ring. In this talk, we study the behavior of the Cox ring of a Mori dream space under modifications, e.g., blow ups. We present an algorithm to compute the Cox ring of the modified variety if possible. As applications, we will study certain Fano varieties. Moreover, we will survey the current range of algorithms for Mori dream spaces.

**27 October 2014: Dmitri Pochekutov (Krasnojarsk)**

**Amoebas of complex hypersurfaces in statistical thermodynamics**

**Abstract:**

The amoeba of a complex hypersurface is its image under the logarithmic projection. A number of properties of algebraic hypersurface amoebas are carried over to the case of transcendental hypersurfaces. We demonstrate the potential that amoebas can bring into statistical physics by considering the problem of energy distribution in a quantum thermodynamic ensemble.

**3 November 2014: Lutz Hille (Münster)**

**Tilting bundles on the pull back of the canonical bundle**

**Abstract:**

In this talk we consider tilting bundles on a smooth projective algebraic variety X and its pull back to the canonical bundle Y. Moreover, we can contract the zero section and obtain an affine Gorenstein singularity Spec R. It turns out, that under certain additional conditions, the tilting bundle is MCM over the ring R. Thus we obtain a diagram X <--- Y ---> Spec R. We start with some simple examples to illustrate the situation, this includes the projective space, the Hirzebruch surfaces and further toric examples. In a second part, we consider a purely algebraic version of this diagram. Finally, we relate this to d-VB-finiteness, a new notion appearing in a joint work with Iyama.

**10 November 2014: Jan Christophersen (Oslo)**

**Local cohomology of the Kähler differentials of invariant rings**

**Abstract:**

Nathan Ilten and I recently proved vanishing results for higher cotangent modules of the Plücker algebra. An important ingredient was the computation of the local cohomology of the Kähler differentials. I will describe this part of our work and relate it to work by Wahl on the cohomology of the square of an ideal sheaf.

**17 November 2014: Christian Gottlieb (Hamburg)**

**Die Verbindung zwischen torischen Degenerationen und Altmanns Deformationen aus Minkowski Zerlegungen**

**Abstract:**

In diesem Vortrag erläutern wir die Verbindung zwischen torischen Degenerationen, die sich aus polyedrischen Zerlegungen ergeben wie sie in der Theorie von M. Gross und B. Siebert behandelt werden und der von K. Altmann diskutierten versellen Deformation, welche aus dem Kegel über einem Polygon entwickelt wird. Eine zentrale Rolle spielt hierbei der Vektorraum, der sich aus den möglichen Minkowski Zerlegungen des Polygons ergibt. Sein Gegenstück findet sich in der entscheidenden Bedingung für wohldefinierte Log-Strukturen auf der zentralen Faser der torischen Degeneration.

**24 November 2014: No talk!**

**1 December 2014: No talk!**

**8 December 2014 **

**15:00!!**

**Giuliano Gagliardi (Tübingen)** and Johannes Hofscheier (Tübingen)

**The generalized Mukai conjecture for symmetric varieties**

**Abstract:**

(1) The Combinatorial Description of Spherical Varieties

A spherical variety is a normal irreducible variety with an action of a connected reductive algebraic group containing an open orbit for a Borel subgroup. Spherical varieties can be considered as a natural generalization of toric varieties. We give a short introduction to the combinatorial description of spherical varieties, which has recently been completed by the work of several authors.

(2) Gorenstein Spherical Fano Varieties

We introduce the combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes (generalizing the reflexive polytopes in the toric case). In this setting, we investigate the generalized Mukai conjecture, an inequality involving the pseudo-index, the dimension, and the Picard number of a smooth Fano variety, leading us to the formulation of a new conjecture involving a certain rational invariant, which will turn out to be of independent interest.

(3) Cox Rings and Spherical Skeletons

We explain how our conjecture may be restated in a purely combinatorial way. This allows us to deduce a (conjectural) combinatorial smoothness criterion for spherical varieties and, finally, to prove our conjecture in the special case of symmetric varieties. In particular, the generalized Mukai conjecture holds in this case.

**15 December 2014: Frederick Witt (Münster) **

**Higgs bundles, Prym varieties and limiting configurations**

**Abstract:**

In this talk we will discuss a (partial) geometric compactification by limiting configurations of the moduli space of Hitchin's self-duality equations. While the proof is in essence analytic we focus on its algebraic interpretation in terms of Higgs bundles and Prym varieties.

**5 January 2015 : No talk!**

**12 January 2015: Elena Martinengo (Hannover)**

**Mori dream stacks**

**Abstract:**

Toric stacks were introduced in a combinatorial way by Borisov-Chen-Smith. Later Fantechi-Mann-Nironi gave a geometric

definition of toric stacks and got a nice classification of them in term of roots over a toric variety.

In a work in collaboration with Andreas Hochenegger, we generalize this work introducing the notion of Mori dream stack. We show that such stacks are preserved under root constructions and taking abelian gerbes. Unlike the case of Mori dream spaces, such stacks are not always given as quotients of the spectrum of their Cox rings by the Picard groups. We give a criterion under which this is true. Finally, we compare this notion with the one of smooth toric stacks.

**19 January 2015: Mateusz Michalek (FU-Berlin, Berkeley)**

**Elementary symmetric polynomials - algebraic geometry inspired by statistics**

**Abstract: **There is a canonical construction in statistics that associates a family of probability distributions to a polynomial. The canonical example is the multivariate Gaussian distribution obtained from the determinant of symmetric matrices. For other hyperbolic polynomials we obtain objects that are interesting from the point of view of statistics, algebraic and convex

geometry.

In my talk I will focus on elementary symmetric polynomials, computing the cohomology class of the graph of the gradient map. This connects to excess intersection theory, toric geometry and Eulerian numbers. The results are a part of a joint work with Uhler, Sturmfels and Zwiernik.

**26 January 2015: Daniel Greb (Essen) **

**Time: 14:15 !!!**

**Movable curves and semistable sheaves**

**Abstract:** I will report on joint work with S. Kebekus and T. Peternell on the one hand and with M. Toma on the other hand, in which we study sheaves that are semistable with respect to movable curves. After motivating why such a generalisation of the classical concept of semistability is natural and desirable, and after discussing the fundamental properties of this notion, I will present an application to higher-dimensional classification theory: every variety with at worst canonical singularities and vanishing first and second Chern class can be obtained as the quotient of an abelian variety by a finite group action.

**2 February 2015: Two talks!!**

**Maria Donten-Bury (FU-Berlin / Warsaw)**

**Time: 14:15 !!!**

**On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32**

**Abstract: **In a joint project with J. Wisniewski we study the symplectic quotient singularity C^4/G where G is a certain matrix group with 32 elements, generated by Dirac matrices. The existence of a symplectic resolution of this singularity was proved by Bellamy and Schedler by non-constructive methods. We give a construction of all its symplectic resolutions using the Cox rings: we determine the Cox ring of a resolution X of C^4/G without knowing any explicit description of X and then we obtain all the sympletic resolutions as GIT quotients of the spectrum of the ring Cox(X). A motivation for this work is a possibility of using the results in the framework of the generalized Kummer construction, which might lead to finding new compact hyperkaehler manifolds.

AND

**Burglind Juhl-Joricke (FU-Berlin)**

**Time: 16:15 !!!**

**Braid invariants and elliptic fiber bundles**

**Abstract:**Take a smooth fiber bundle over an open Riemann surface whose fibers are tori (elliptic fiber bundle). We consider the question whether the bundle is isotopic to a holomorphic bundle. When the open Riemann surface is an annulus this question can be answered in terms of a single invariant, the conformal module of the bundle. Moreover, the isotopy class of the bundle is determined by an isotopy class of self-homeomorphisms of a fiber (more precisely by its conjugacy class). It turns out that the conformal module of the bundle is inverse proportional to the entropy of the conjugacy class.

The question for elliptic fiber bundles over a torus with a hole will also be discussed.