# SoSe2015

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

# Schedule:

**27 April 2015: Winfried Bruns (Osnabrück)**

**Maximal minors and linear powers**

**Abstract: **We say that an ideal I in a polynomial ring S has linear powers if all the powers of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that all ideals defining rational normal scroll have linear powers.

Joint work with Aldo Conca and Matteo Varbarao (J. Reine Angew. Math., to appear).

**4 May 2015: Matej Filip (FU-Berlin)**

**Deformation Quantization of Toric Varieties**

**Abstract:**We give review of the non-commutative deformation theory in terms of the Hochschild cochain complex and quantization of Poisson structures with Kontsevich's formality theorem in the smooth setting. We analyse in more detail the case of singular toric varieties.

**11 May 2015: Mina Bigdeli, (Zanjan)**

**Simplicial Complexes of Whisker Type**

**Abstract: **

Graphs with whiskers have first been considered by R. H. Villarreal. They all share the nice property that they are Cohen-Macaulay. Various extensions of this concept and generalizations of his result have been considered in the literature. Here we consider the following variation: Let I ⊂ S=K[x1, . . . ,xn] be a zero-dimensional monomial ideal. In analogy to the classical case, the simplicial complex \Theta(I), whose facet ideal coincides with the polarization I^℘ of I, we call of whisker type – the whiskers being the simplices corresponding to the polarization of the pure powers contained in I.

It is shown that the independence complex of \Theta(I), denoted \Delta(I), is vertex decomposable, and hence shellable. Furthermore, it is shown that all powers of the of the Alexander dual L(I) of the polarization of I have linear quotients and an explicit formula of the depth functionf(k)= lim_{k→∞} depth S/L(I)^k is given.

Joint work with Jürgen Herzog, Takayuki Hibi and Antonio Macchia (Electronic J . Comb., to appear).

**18 May 2015: Andreas Hochenegger (Köln):**

**14:15!!**

**Maps between Mori Dream spacesAbstract: **

Maps between toric varieties can be lifted to maps between their Cox rings, if one is allowed to modify the Cox ring of the domain a bit. This was shown by Gavin Brown and Jaroslaw Buczyński. In a work in progress with Elena Martinengo, we note that this idea becomes more natural when considering quotient stacks. We also extend this lifting to maps between Mori Dream spaces. In my talk, I present these ideas and further directions.

**AND**

**16:15**

**Elena Martinengo (Hannover):**

**Singularities of moduli spaces of sheaves on K3 surfaces**

**Abstract:**

In the eighties Mukai proved that the singularities of the moduli space of sheaves on a K3 surface are contained in the locus of strictly semistable sheaves, that is not empty just if the polarization is non generic or if the Mukai vector is non primitive. In the first case, Kaledin, Lehn and Sorger conjectured that the dg-algebra that controls deformations of sheaves on a K3 is formal. This would give a complete description of the singularities of the

moduli space.

The conjecture was proved in some cases by Kaledin-Lehn and Zhang. The tecniques they used are similar and they consist in pulling back the sheaves on the K3 to the twistor family and to apply Kaledin's theorem of formality in families.

In a work in progress with Manfred Lehn we aim to complete the proof of the conjecture. We proved the conjecture of the remaining case and we are trying to extend our ad hoc construction to a general proof.

**25 May 2015: Holiday--no talk**

**1 June 2015: David Ploog (Bonn)**

**Spherical subcategories**

**Abstract: **

Spherical objects are important sources of derived symmetries, they are studied for varieties, algebras, and beyond. In this talk, we pursue the question if objects satisfying a weaker condition (we drop the Calabi-Yau condition) still have some intrinsic meaning. The answer is yes, and among other applications, it leads to a new invariant of triangulated categories.

Examples from algebraic geometry and representation theory will be given. (Joint work with Andreas Hochenegger and Martin Kalck.)

**8 June 2015: Alex Küronya (Frankfurt)**

**Newton-Okounkov bodies with applications to diophantine questions ** **and local positivity**

**Abstract:**

The concept of Newton-Okounkov bodies is a recent attempt to understand the asymptotics of vanishing behaviour of global sections of line bundles on projective varieties via convex geometry. In this talk we first give a quick outline of the existing theory along with the construction of geometrically significant concave functions on Okounkov bodies, which then leads to a recent application due to McKinnon and Roth in diophantine geometry. The last part of the lecture will be devoted to a study of local positivity of ample line bundles via Okounkov bodies, a joint work with Victor Lozovanu.

**22 June 2015: Laura Matusevich (Texas A&M)**

**TALK CANCELLED!**

**29 June 2015: **

**14:15!!**

**Lutz Hille (Münster)**

**Derived equivalences for various weighted projective spaces and stacks**

**Abstract:**

There are weighted projective spaces in the classical sense, stacky weighted projective spaces, and weighted projective spaces in the sense of Geigle and Lenzing. We compare these constructions via morphisms and derived categories. It turns out that the weighted projective space as a stack is derived equivalent to any crepant resolution of the classical one, if it is Fano.

We remind on the constructions, review the Fano case and the resolutions. Moreover, we explain the classical example by Ballard, Favero, Katzarkov: the derived category of the second Hirzebruch surface is equivalent to the weighted projective stack P(2,1,1). The main result generalizes this construction forany Fano weighted projective space.

Eventually, we give several applications and motivations for such a result.

**AND**

**16:15!!**

**Carsten Lange (München)**

**Generalized permutahedra and Minkowski decompositions**

**Abstract:**

Generalized permutahedra are a family of convex polytopes with combinatorial and geometric properties imposed by the symmetric group. They can be realized as lattice polytopes and describe resolutions of singularities of toric varieties associated to the symmetric group in generic situations. In my talk I will explain how to obtain a unique Minkowski decomposition into faces of a standard simplex for these polytopes and I will show by example that the combinatorial type of these polytopes does not determine this decomposition uniquely. Finally, I will discuss the possibility of other Minkowski decompositions of generalized permutahedra.** **

**6 July 2015: Konstantinos Georgiadis**