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# SoSe 2011

Seminar: Algebraic Geometry Summer Semester 2011

Unless otherwise specified, all talks take place in room 119, Arnimallee 3 and begin at 16:00.

Schedule:

 11.04.2011 Lars Petersen (FU-Berlin) Okounkov bodies of complexity-one T-varieties Abstract: In recent years, so-called (Newton-)Okounkov bodies have seen a considerable rise in popularity. In this talk, I will give a gentle introduction to this beautiful theory which entangles algebraic and convex geometric notions. Finally, I will report on new results regarding the computation of Okounkov bodies for projective complexity-one T-varieties. 18.04.2011 Andreas Hochenegger (FU-Berlin) Constructing exceptional sequences on toric varieties Abstract: In Kawamata's article "The derived category of toric varieties", he constructs exceptional sequences (ie. a special generator set of the derived category) for toric varieties. This can be done inductively by using the toric Minimal Model Program. Actually, Kawamata's method needs a generalisation of varieties to so-called orbifolds - a special form of stacks. In this talk I want to give an overview of this article. I will focus on the point, why orbifolds enter the picture, and how they can be described in the toric setting. 25.04.2011 Ostermontag! 02.05.2011 Hélène Esnault (Essen) Entropy on surfaces in characteristic p>0 Abstract: We show that the value of the entropy on $\ell$-adic cohomology of an autmorphism of a smooth projective surface over a finite field is taken on the N\'eron- Severi group. Over the field of complex numbers, this is a consequence of Hodge theory. Over a finite field, this would be a consequence of the standard conjectures. Joint work with V. Srinivas. 09.05.2011 Bernd Sturmfels (UC Berkeley and Matheon Berlin) Mustafin Varieties Abstract: A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in the Bruhat-Tits tree of PGL(2) and in the two-dimensional building of PGL(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types. This is joint work with Dustin Cartwright. Mathias Haebich and Annette Werner (arXiv:1002.1418). 16.05.2011 Lars Kastner (FU Berlin) Calculating Generators of Multigraded Algebras Abstract: I will report on the following result from joint work with Nathan Ilten: Every integral, normal, finitely generated C-algebra graded by a lattice M arises from a so-called p-divisor on a semiprojective normal variety. We will describe an algorithm to compute the generators of such an algebra given by a p-divisor on a semiprojective normal variety. 23.05.2011 Alastair Craw (Glasgow) Mori Dream Spaces and multigraded linear series Abstract: I'll describe recent joint work with my student Dorothy Winn that constructs Mori Dream Spaces as fine moduli spaces of quiver representations, extending results from Craw-Smith for projective toric varieties. Our approach embeds a Mori Dream Space into the "multigraded linear series" of a collection of line bundles, generalising the classical construction of the linear series of a single line bundle. 30.05.2011 Thomas Bauer (Marburg) Seshadri constants and the generation of jets Abstract: In this talk I will report on work revolving around the connection between Seshadri constants and the generation of jets. It is well-known that one way to view Seshadri constants is to consider them as measuring the rate of growth of the number of jets that multiples of a line bundle generate. Here we ask, conversely, what we can say about the number of jets once the Seshadri constant is known. As an application of our results, we prove a characterization of projective space among all Fano varieties in terms of Seshadri constants. 06.06.2011 Marianne Merz (FU-Berlin) Persistent homology Abstract: Persistent homology is an algebraic tool for measuring topological features of shapes and functions. I will give a short history of persistence and present its basic concepts as well as an algorithm for computing. 13.06.2011 Pfinstmontag! 20.06.2011 Rita Pardini (Pisa) Curves on irregular surfaces and Brill-Nother theory Abstract: The irregularity of a smooth complex projective surface is the number q of independent global 1-forms of S; there exist a complex torus of dimension q, the Albanese variety Alb(S), and a map S-->Alb(S), the Albanese map, through which any map S-->T, T a complex torus, factorizes. The Albanese dimension of a surface is the dimension of the image of the Albanese map. Little is known on surfaces of general type with Albanese dimension 2. I will propose an approach to the study of these surfaces via the analysis of the curves of small genus on them. This leads naturally to considering the Brill-Noether locus W(C) of a curve C of S, namely the set of line bundles P in Pic^0(S) such that the divisor C+P is effective. I will give a structure result for W(C) and show that it gives numerical restrictions on the curves of small genus on S. This is joint work with Margarida Mendes Lopes and Gian Pietro Pirola. 27.06.2011 Jarek Wisniewski (Warsaw) Differentials of Cox rings: Jaczewski's theorem revisited Abstract: The Cox ring of a (complex) projective variety provides information about the geometry of the variety and its small birational modifications. The module of its differentials can be described in terms of the Euler sequence on the variety, or the universal Atiyah extension. The situation is particularly nice if the variety in question is toric. I will introduce these notions and explain the results of a joint work with Oskar Kedzierski. 04.07.2011 Georg Hein (Esssen) Thetareihen quadratischer Formen Abstract: Die Thetareihe einer quadratischen Form ist eine Invariante, die es erlaubt solche Formen zu unterscheiden. Da im Falle ganzzahliger Formen diese Thetareihen Modulformen sind, kann man diese sehr effektiv berechnen. Ich möchte eine Konstruktion von isospektralen quadratischen Formen praesentieren, die eine Konstruktion von Schiemann, Conway & Sloane verallgemeinert. Abschliessend möchte ich neue Invarianten vorstellen, die diese Formen unterscheiden können. 11.07.2011 Thomas Peternell (Bayreuth) Singular varieties with trivial canonical classes. Abstract: Der klassische Satz von Beauville-Bogomolov besagt, dass sich jede projektive oder kählersche komplexe Mannigfaltigkeit mit trivialer kanonischer Klasse in Tori, Calabi-Yau und hyperkähler-Mannigfaltigkeiten zerlegen lässt. Diese Varietäten haben alle Kodaira-Dimension 0. Um allgemeiner Varietäten mit Kodaira-Dimension 0 zu verstehen, ist es jedoch notwendig Varietäten mit sog. kanonischen Singularitäten zu studieren, deren kanonische Klasse trivial ist. Solche singuläre Varietäten werden in dem Vortrag eingehend diskutiert. 16.09.2011 Ivan Cheltsov (Edinburgh) Cremona group and the icosahedron Abstract: We discuss the relation between the group of rotation of the icosahedron and the groups of birational automorphisms of projective plane and projectie space.