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

Forschungsseminar "Algebraische Geometrie"

Sommersemester 2014

Das Forschungsseminar findet mittwochs in der Zeit von 15:00 - 17:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 2.009 (Haus 2, Erdgeschoss), statt.

Seminar: Algebraic Geometry an der FU

16.04.2014 Alessandro Verra (Università Roma Tre)
Title: Six-dimensional Pryms and Fano-Enriques threefolds

Abstract: The talk surveys some natural relations between the moduli spaces of Prym varieties and that of Enriques surfaces in genus at most 6. Then new relations are presented between R7 and the family of Fano-Enriques threefolds which are linear sections of the variety of rank two quadrics in the three-dimensional projective space. This leads to a geometric description of the universal singular locus of the Prym theta divisor over R7 and to a proof of its rational connectedness.
23.04.2014 Gavril Farkas (HU Berlin)
Title: What is the general principally polarized abelian variety of dimension six?

Abstract: The general principally polarized abelian variety of dimension at most five is known to be a Prym variety. This reduces the study of abelian varieties of small dimension to the beautifully concrete theory of algebraic curves. I will discuss recent progress on finding a structure theorem for principally polarized abelian varieties of dimension six, and the implications this uniformization result has on the geometry of the moduli space A6.
30.04.2014 kein Seminar
07.05.2014 kein Seminar
14.05.2014 Michael Kemeny (HU Berlin)
Title: Singular Curves on K3 Surfaces

Abstract: In this talk we discuss the moduli space of smooth curves with a nodal model on a K3 surface. We will show that this space has the maximal possible dimension in most cases, as well as attempts to describe it via Brill-Noether theory and Gaussian-type conditions.
21.05.2014 Alina Marian (Northeastern University)
Title: Geometry of the Verlinde sheaves

Abstract: The Verlinde sheaves on the moduli space of curves arise by considering relative moduli spaces of vector bundles over Mg. They can also be constructed on the moduli spaces of K3 and abelian surfaces. I will review the construction and will describe recent progress understanding the geometry in all three settings - curves, abelian and K3 surfaces - with an emphasis on the case of curves.
28.05.2014 Juan Carlos Naranjo (Universitat de Barcelona)
Title: Isogenies of Jacobians

Abstract: We prove by means of the study of the infinitesimal variation of Hodge structure and a generalization of the classical Babbage-Enriques-Petri theorem that the Jacobian variety of a generic element of a k codimensional subvariety of Mg is not isogenous to a distinct Jacobian if g>3k+4. We extend this result to k=1, g\ge 5 by using degeneration methods. This is a joint work with G.P. Pirola and V. Marcucci.
16:30 Uhr Charles Siegel (Kavli IPMU, Japan)
Title: Trees and an Affine Cover of M0,n+1

Abstract: I will talk about work in progress, with Satoshi Kondo (Kavli IPMU) and Jesse Wolfson (Northwestern), on a covering of of the moduli of genus 0 marked curves by open affine subvarieties determined by rooted trees and constructed via Kapranov's model in terms of the blow up of projective space along linear subvarieties.
04.06.2014 Yohan Brunebarbe (MPI Bonn)
Title: Symmetric differentials and variations of Hodge structures

Abstract: Let U be a smooth algebraic complex variety and X be a compactification such that the complementary D = X - U has simple normal crossings. In my talk I will explain how the existence on U of a non-trivial Z-polarized variation of Hodge structures forces the existence of non-zero logarithmic symmetric differential forms on the pair (X,D), i.e. sections of symmetric powers of the logarithmic cotangent bundle of the pair (X,D). This extends to the non-compact case one of the main result of a previous joint work with B. Klingler and B. Totaro. I will also discuss some applications to smooth quotients of bounded symmetric domains by arithmetic groups and to moduli spaces of curves and polarized K3 surfaces.
11.06.2014 kein Seminar
18.06.2014 Sebastian Casalaina-Martin (University of Colorado, Boulder)
Title: Prym varieties and applications

Abstract: Prym varieties are abelian varieties obtained from connected, etale double covers of curves. They were introduced by Mumford (1974) in connection with the study the intermediate Jacobian of a cubic threefold, and were later used by Beauville (1977) to extended Mumford's construction to the case of arbitrary fibrations in odd dimensional quadrics. In this talk, I will review these constructions and then discuss a few recent projects building on this work. This will include degenerations of intermediate Jacobians, extensions of the Prym map, and the cohomology of fibrations in quadrics over the rational numbers. This is joint work with (in various combinations) S. Grushevsky, K. Hulek, R. Laza and J. Achter.
25.06.2014 Jesse Kass (University of Michigan)
Title: What is the limit of a line bundle on a nonnormal variety?

Abstract: On a nonnormal variety, the limit of a family of line bundles is not always a line bundle. What is the limit? I will present an answer to this question and give some applications. If time permits, I will discuss connections with autoduality for the compactified Jacobian and with recent divisor class group computations of R. Hartshorne and C. Polini.
02.07.2014 kein Seminar
09.07.2014 Balázs Szendrői (University of Oxford)
Title: Euler characteristics of Hilbert schemes of points of ADE surfaces

Abstract: Given a smooth surface, the generating series of Euler characteristics of its Hilbert schemes of points can be given in closed form by (a specialisation of) Goettsche's formula. I will discuss a generalisation of this formula to surfaces with rational double points, built from the representation theory of affine Lie algebras. (Joint work with Adam Gyenge, Budapest)
16.07.2014 kein Seminar


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