# Sommersemster 2016

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

**25th April 2016 **

**David Ploog (FU-Berlin)**

**Rigid and spherelike divisors on surfaces****Abstract:** We look at effective divisors on surfaces with interesting numerical, intersection, and cohomological behaviour. The simplest kind of example are trees of -2-curves, which include the exceptional divisors of ADE singularities. This picture extends nicely to other negative divisors. There will be many examples. Joint work with Andreas Hochenegger.**2nd May 2016 **

**Lars Kastner (FU-Berlin) **

**Toroidal varieties**

**9th May 2016**

**Lars Kastner (FU-Berlin) **

**Semi stable reduction**

**16th May 2016: Pentecost**

**23rd May 2016**

**Vijaylaxmi Trivedi (Mumbai)**

**Hilbert-Kunz density function and Hilbert-Kunz multiplicity**

**Abstract: **In this talk we recall a well-studied char p invariant, the Hilbert-Kunz multiplicity, for a pair (R,I), where R is a local ring/standard graded ring and I is an ideal/graded ideal of finite colength. This could be considered as an analogue of the Hilbert-Samuel multiplicity (but specific to characteristic p > 0).

We give a brief survey of some of the results on this invariant and try to convey why the HK multiplicity is a better and a worse invariant than Hilbert-Samuel multiplicity of a ring.

In the graded case (based on the recent work), for a pair (R, I) we introduce a new invariant, the Hilbert-Kunz density function, which is a limit of a uniformly convergent sequence of real valued compactly supported, piecewise linear and continous functions. We express the HK multiplicity as an integral of this function.

We prove that this function (unlike the HK multiplicity) satisfies a multiplicative formula for the Segre product of rings (the homogeneous coordinate ring of the Segre product of projective varieties). As a consequence some known results for the HK multiplicity of certain rings also hold for the HK multiplicity of their Segre products. This leads to some new computations of the HK multiplicity.

We discuss a few other applications of this function, like asymptotic behaviour of the HK multiplicity of R with repect to the powers of I and a possible approach for the HK multiplicity in characteristic 0 through reduction mod p.

**30th May 2016**

**Helge Ruddat (Mainz)**

**Block-Göttsche type invariants in higher dimensions**

**Abstract:** Tropical curve counts in toric varieties match up with log Gromov-Witten invariants when imposing a suitable set of incidence and psi class conditions (assuming non-superabundancy). I prove this generalization of Mikhalkin's and Siebert-Nishinou's results in a joint work with Travis Mandel. I will discuss the tropical version of a pencil of cubics in the projective plane, how this helps to find the rational curves in the pencil and then relate this to real curve counts via Welschinger invariants. I will explain q-deformed Gromov-Witten invariants and introduce such in higher dimensions conjecturing and proving some novel results in that direction. If time permits, I mention applications to mirror symmetry.

**6th June 2016**

**Pawel Sosna (Hamburg)**

**On the derived category of the Hilbert scheme of two points**

**Abstract: **The Hilbert scheme of n points on a smooth projective surface is known to be smooth. A theorem of Bridgeland-King-Reid, Haiman identifies the bounded derived category of coherent sheaves on the Hilbert scheme of n points with the bounded derived category of equivariant sheaves on the n-fold product of the surface. On the other hand, the Hilbert scheme of two or three points on any smooth projective variety X is also smooth. In this talk I will talk about the structure of the bounded derived category of the Hilbert scheme of two points on X. This is joint work in progress with Andreas Krug.

**20th June 2016**

**Bernd Sturmfels (Berkeley & TU-Berlin)**

**Khovanskii Bases**

**Abstract: **Given an algebra generated by polynomials over a field with valuation, its Khovanskii basis has the property that the initial forms generate the initial algebra. This generalizes the familiar notion of SAGBI bases, which relies on term orders, and it offers a computational framework for toric degenerations and Newton-Okounkov bodies. This lecture provides an introduction and an application to Cox rings of del Pezzo surfaces.

**27th June 2016**

**Lutz Hille (Münster)**

**Curves with with negative selfintersection and derived categories, jt. withDavid Ploog**

**Abstract: **Recently we started to study certain subcategories of the derived category of coherent sheaves on a surfaces, allowing to get a better understanding of spherical objects and autoequivalences of the derived category. From the geometric side we are interested in surfaces and like to classify exceptional and spherical objects in the category of coherent sheaves and its derived category. On the algebraic side we are interested in algebras of global dimension two and the analogous problem. In this talk we associate to a A_n-configuration of curves with negative selfintersection in a surface X a subcategory of coherent sheaves that is equivalent to a module category over a finite dimensional algebra. Then we classify for (-2)-curves all exceptional and spherical modules over this algebra, that in turn classifies all exceptional and spherical coherent sheaves.

We give an overview on the general results, we consider the particular case of(-2)-curves in detail, and finally give a list of open problems.

**4th July 2016**

**Ana Maria Botero (HU-Berlin)**

**Intersection theory of b-divisors on toric varieties**

**Abstract: **We introduce toric b-divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions, toric b-divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric b-divisor corresponds to the number of lattice points in this convex set and we give a Hilbert-Samuel type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. We further investigate the question of extending these results to arbitrary toroidal varieties. Examples in which such b-divisors naturally appear are invariant metrics on line bundles over toroidal compactifications of mixed Shimura varieties. Indeed, the singularity type which the metric acquires along the boundary can be encoded using toroidal b-divisors.

**11th July 2016**

**Peter Schenzel (Halle)On the visualization of blow-ups of the plane in points**

**Abstract: **In the talk we investigate the blowing up of a set of finite points in the affine plane. It will be shown how to use these methods in order to visualize blowing ups of them by an embedding in a torus with a rational parametrization (as Brodmann suggested). We derive an implicit surface that allows to visualize the toroidal blowup as well as its deformations

by several parameters interactively in real-time (using the program RealSurf developed by C. Stussak). The methods provide insights in the structure of blowups of points, even if the points are interactively moved or the definig equations degenerate.