Humboldt Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik

Forschungsseminar "Algebraische Geometrie"

Sommersemester 2015


Das Forschungsseminar findet mittwochs in der Zeit von 15:00 - 17:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 2.006 (Haus 2, Erdgeschoss), statt.

Seminar: Algebraic Geometry an der FU



A classical theorem of Severi classifies smooth, integral surfaces X in P^5 such that the family of their bisecant lines has order 1, that is exactly one line of the family is passing through a general point of P^5. Natural generalizations of this theorem are described, in order to answer the following question: when the congruence of k-secant r-spaces to a projective integral variety in P^n has order 1? The problem and its answer are studied in the next case to be considered: namely the congruence of 4-secant planes to a surface in P^6.
15.04.2015 Andreas Leopold Knutsen (University of Bergen)
Title: Moduli of nodal curves on K3 and abelian surfaces

Abstract: The study of nodal curves in the (complex) plane was initiated by Severi in the 20s. Results of Severi, Harris and Sernesi imply that the variety parametrizing plane nodal curves of any fixed degree and allowed geometric genus (so-called Severi varieties) is nonempty, smooth and irreducible of the expected dimension, and the family of curves has the expected number of moduli (meaning that their normalizations have the expected number of moduli). Similar questions are quite open on most other surfaces. In the talk I will present recent progress concerning the moduli problem on K3 and abelian surfaces, where the natural objects to consider are the varieties parametrizing nodal curves on the surfaces allowing the surfaces to move in suitable moduli spaces (so-called universal Severi varieties). It turns out that such families of curves have the expected number of moduli except possibly for finitely many cases (and two known exceptions in the case of smooth curves). The K3 case is work joint with Ciliberto, Flamini and Galati, and partially intersects recent results obtained by Kemeny. The abelian case is joint work with Lelli-Chiesa and Mongardi.
23.04.2015 (DONNERSTAG, RAUM 4.007) Rahul Pandharipande (ETH Zürich) und Dragos Oprea (University of California, San Diego)
Title Rahul Pandharipande: A compactification of the space of meromorphic differentials

Abstract: I will discuss a proposal for a compactification of the spaces of (C,w) where C is a nonsingular curve and w is a meromorphic differential with specified orders of zeros and poles. Joint work with G. Farkas.
Title Dragos Oprea: The Chern characters of the Verlinde bundles

Abstract: The Verlinde bundles are constructed over the moduli space of curves by considering relative moduli spaces of vector bundles. Their Chern characters yield a cohomological field theory. I will explain how Teleman's work on semisimple cohomological field theories can be used to derive explicit formulas for the Chern characters in terms of tautological classes. This is joint work with A. Marian, R. Pandharipande, A. Pixton and D. Zvonkine.
29.04.2015 Frank Gounelas (HU Berlin)
Title: Positivity of the cotangent bundle of Calabi-Yau varieties

Abstract: In this talk I will discuss various notions of positivity for a vector bundle and how these are related to classification problems of higher dimensional algebraic varieties in the case of the cotangent bundle. Studying in particular the case of Calabi-Yau varieties of dimension two and three, which lie very close to the border between varieties of non-negative and negative Kodaira dimension, one finds a rich source of interesting examples.
06.05.2015 Alex Küronya (Goethe-Universität Frankfurt am Main)
Title: Newton-Okounkov bodies 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. The second part of the lecture will be devoted to a study of local positivity of ample line bundles via Okounkov bodies.
13.05.2015 Takehiko Yasuda (Osaka University, zzt. Max-Planck-Institut für Mathematik)
Title: The wild McKay correspondence and stringy invariants

Abstract: A version of the McKay correspondence in terms of stringy invariants was proved by Batyrev. Later, Denef and Loeser found an elegant approach to this result by using motivic integration. In this talk, we discuss a generalization of their works to positive or mixed characteristic, emphasizing the case where the given finite group has order divisible by the characteristic of the base/residue field. To explain the main result, let us consider a finite group G acting on an affine space V over the integer ring O of a local field K (for instance, K=k((t)) and O=k[[t]] with k a finite field). Suppose there exists a crepant resolution Y of V/G. The main result roughly says that the number of rational points of Y over the residue field of O is equal to a weighted count of G-extensions of K. Such weighted counts of G-extensions had been originally studied in the number theory. A special case of our result relates Bhargava’s mass formula of etale algebras over a local field with the Hilbert scheme of points on the affine plane.
20.05.2015 kein Seminar
28.05.2015 (DONNERSTAG) Alessandro Verra (Università Roma Tre)
Title: Congruences of order 1 of secant spaces to a projective surface

Abstract: A classical theorem of Severi classifies smooth, integral surfaces X in P5 such that the family of their bisecant lines has order 1, that is exactly one line of the family is passing through a general point of P5. Natural generalizations of this theorem are described, in order to answer the following question: when the congruence of k-secant r-spaces to a projective integral variety in Pn has order 1? The problem and its answer are studied in the next case to be considered: namely the congruence of 4-secant planes to a surface in P6.
03.06.2015 kein Seminar
10.06.2015 Victoria Hoskins (FU Berlin)
Title: To be announced

Abstract: To be announced.
17.06.2015 Nicola Tarasca (University of Utah)
Title: To be announced

Abstract: To be announced.

--------------------------------------------------------------------------------------
Archiv:

Wintersemester 2014/15

Sommersemester 2014

Wintersemester 2013/14

Sommersemester 2013

Wintersemester 2012/13

Sommersemester 2012

Wintersemester 2011/12

Sommersemester 2011

Wintersemester 2010/11