# WS2016/17

**Unless otherwise specified, all talks take place in room 119, Arnimallee 3 and begin at 16:15.****Schedule: ****SFB-Seminars on 25th October 2016:**

**Michael Wemyss (Glasgow): ****The Homological MMP**

**10:00, Arnimallee 3, SR 005**

I will explain how, given a crepant morphism with one-dimensional fibres between 3-folds, it is possible to use noncommutative deformations to jump between minimal models in a satisfyingly algorithmic fashion. As part of this, a flop is viewed homologically as the solution to a universal property, and so is constructed not by changing GIT, but

instead by changing the algebra in a cluster-like way. Carrying this extra information allows us to iterate, without having to

calculate everything from scratch. Proving things in this way has many other consequences, and I will explain some of them, both theoretical and computational.

**PD. Dr. David Ploog (FU Berlin)****Classical deformation theory**

**16:00, Königin-Luise-Str. 12-16**

**Abstract:** The concept of deforming a mathematical object, such as a variety, a bundle on a manifold, an algebra, or a complex structure, is very old. In many cases, infinitesimal deformations can be understood very well through homological methods.

In the first talk, we will give a gentle introduction to this topic, and explain in concrete examples the concept of functors on Artin rings. These capture the concept of infinitesimal deformations in a crucial way.

**Dr. Will Donovan (Kavli IPMU)**

** Non-commutative deformations and flops**

**17:00, Königin-Luise-Str. 12-16**

**Abstract: **The second talk presents recent research by Will Donovan and Michael Wemyss: the idea is to allow non-commutative rings as bases for deformations. This has surprising applications and allows, for example, to understand geometric problems about flops that have been open for decades.

**Seminar talks:**

**31st October 2016**

**Prof. Frederik Witt (Stuttgart)**

**Algebraic geometry and Riemannian holonomy**

**Abstract:** In this survey talk I will explain the link between Calabi-Yau and irreducible symplectic/hyperkähler manifolds in complex-algebraic geometry and Ricci-flat manifolds of special holonomy in Riemannian geometry following Beauville's excellent survey arXiv:math.ag/9902110.

**7th November 2016**

**Kathlén Kohn (TU Berlin + Berkeley) will give a talk on**

**Coisotropic Hypersurfaces in Grassmannians**

**Abstract:** For every variety X in projective n-space of dimension k, the set ofall projective subspaces of dimension n-k-1 that intersect X is a hypersurface in the Grassmannian of these subspaces. This hypersurfaceis defined by a polynomial in the Plücker coordinates of this Grassmannian, which is unique up to scaling and the Plücker relations. This polynomial is called the Chow form of X. One can generalize this definition to projective subspaces of higher dimension: All subspaces of a fixed dimension that intersect X non-transversally form a subvariety of a Grassmannian, which is said to be coisotropic. We will study which coisotropic subvarieties are hypersurfaces, and we will show that the degrees of the coisotropic hypersurfaces are the well-studied polar degrees. Moreover, the coisotropic hypersurfaces of X and its projectively dual variety will be related, and we will see that all hyperdeterminants arise as coisotropic forms of Segre varieties. Finally, we will investigate how to recover the underlying projective variety X from a given coisotropic hypersurface and how to test if a given subvariety of a Grassmannian is coisotropic independently of X.

**14th November 2016 **um 14:15 !!!

**Jaroslaw Wisniewski (Warsaw)**

**Flag varieties, a geometric characterization and rigidity**

**Abstract: **Flag varieties can be characterized as these Fano manifolds whose all extremal contractions are smooth P^1 fibrations. Subsequently they are (globally) rigid in families of Fano manifolds. I will report on these results obtained in collaboration with Gianluca Occhetta, Luis E. Solá Conde, Kiwamu Watanabe, and Andrzej Weber.

**21st November 2016**

**Jan Christophersen (Oslo)**

**Derivations of artinian algebras and rigidity of curve singularities**

**Abstract: **Let A be a local artinian algebra over an algebraically closed field k of characteristic 0. There is a conjecture that the vector space dimension of the derivations of A is greater or equal the vector space dimension of the maximal ideal of A. Moreover equality should only happen when A is a hypersurface. We will show that this conjecture would imply non-rigidity of reduced curve singularities and describe various attempts to prove the conjecture. In particular we relate the problem to the GL(n) representation on n commuting n x n matrices.

**5th December 2016**

**Joshua Jackson (Oxford)**

**Variation of Non-reductive GIT Quotients via the U-hat Theorem**

**Abstract: **One of the salient features of Geometric Invariant Theory is the dependence of the quotient on an additional parameter, a so-called choice of linearisation. It has long been known that the parameter space has a 'wall-and-chamber' structure, leading to wall-crossing phenomena which give different interrelated birational models of the quotient. After giving a brief introduction to the classical, i.e. reductive, GIT picture, I will talk about recent developments in non-reductive GIT, explaining how the 'U-hat' theorem allows us, in favourable conditions, to recover many of the good properties enjoyed in the reductive setting

**12th December 2016**

**Lattice Polytope Workshop (C.Haase) => no talk**

**9th January**

**Georg Hein (Essen)**

**Theta Divisors for Holomorphic Triples**

**Abstract: **We show how to extend the theory of the generalized theta divisor to the setting of holomorphic triples. This gives an effective base point free divisor on this coarse moduli space possessing a geometric interpretation.

**23rd January**

**Jaroslaw Buczyński (Warsaw)**

** Maps of Mori Dream Spaces**

**Abstract:** Any rational map between affine spaces or projective spaces can be described in terms of their (homogeneous) coordinates. Toric Varieties and Mori Dream Spaces are classes of algebraic varieties for which there exists a sensible analogue of homogeneous coordinate ring. I will present how to obtain a description of a rational map of Mori Dream Spaces (or Toric Varieties) in terms of such coordinate rings. More precisely (in the case of regular maps) I will show there exists a finite extension of the coordinate ring of the source, such that the regular map lifts to a morphism from the Cox ring of the target to the finite extension. Moreover the extension only involves roots of homogeneous elements. Such a description of the map can be applied in practical computations.

The talk is based on my joint works with Gavin Brown and Oskar Kedzieski, and also on the master thesis of Tomasz Mandziuk.