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

Forschungsseminar "Algebraische Geometrie"

Wintersemester 2016/17

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

19.10.2016 kein Seminar
26.10.2016 Hsueh-Yung Lin (HU Berlin)
Title: Bimeromorphic Kodaira problem for non-uniruled Kähler 3-folds

Abstract: The bimeromorphic Kodaira problem asks whether a compact Kähler manifold has a bimeromorphic model which is deformation equivalent to a projective variety. After giving a survey of known results and introducing Kähler 3-folds from the point of view of the minimal model program, we will focus on compact Kähler 3-folds of Kodaira dimension 1, and show that the bimeromorphic Kodaira problem has a positive answer for these manifolds. Together with earlier work of N. Nakayama and recent work of P. Graf, we answer positively the bimeromorphic Kodaira problem for non-uniruled Kähler 3-folds, as predicted by a conjecture of T. Peternell.
02.11.2016 Marian Aprodu (U Bucharest)
Title: Syzygies and secant loci

Abstract: We discuss the interactions between syzygies of curves and the geometry of the secant loci in the symmetric products. We show that a regular behavior of these loci for special line bundles imply the vanishing of linear syzygies. The talk is based on a joint work with Edoardo Sernesi.
16:00 Uhr s.t.
Alessandro Verra (U Roma 3)
Title: Rational parametrizations of universal K3 surfaces of low genus via cubic fourfolds.

Abstract: In complex projective geometry the interplay between K3 surfaces and cubic fourfolds is a well known topic, related to the rationality problem for these fourfolds. In the talk a survey on the topic and its recent results is given. Then special cases of interest are studied. One is the family of conjecturally rational cubic fourfolds X containing a 3-nodal rational scroll R of degree 7. This family is associated to the moduli F 14 of genus 14 K3 surfaces. The following new result is presented: the moduli space of pairs (X, R) is rational and birational to the universal K3 over F 14. The method applies to the next case to be considered, namely the space of moduli of pairs (X, R) such that R is an 8-nodal rational scroll. In this case the unirationality of this space, and hence of the universal K3 surface of genus 42, can follow.
16.11.2016 Rahul Pandharipande (ETH Zürich)
Title: The r-spin CohFT for higher r

Abstract: I will discuss the Cohomological Field Theory obtained from the r-spin class on the moduli space of r-spin curves for higher r. Exact calculations can be made at particular semisimple points. The results include exact formulas for r-spin correlators in genus 0, useful families of tautological relations in higher genus, and a surprising connection to the moduli of holomorphic differentials. The talk represents joint work with F. Janda, A. Pixton, and D. Zvonkine.
23.11.2016 Kay Rülling (HU Berlin)
Title: Correspondence action on Witt vector cohomology

Abstract: This is a report on joint work with Andre Chatzistamatiou. I will explain Witt vectors, the de Rham-Witt complex and how correspondences act on their cohomology. Then I will explain how to use this to show that Witt vector cohomology is a birational invariant on smooth projective varieties in positive characteristic and define Witt rational singularities.
30.11.2016 (Erwin-Schrödinger-Zentrum u. Johann von Neumann-Haus) Joint seminar in algebraic geometry with the Sturmfels group from Leipzig/Berlin
13:30 - 14:30 Bernd Sturmfels (U Berkeley/MPI MIS Leipzig), Raum 0'101 (Erwin-Schrödinger-Zentrum)
Title: Real Rank Two Geometry

Abstract: The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two. Joint with Anna Seigal.
15:15 - 16:15 Igor Dolgachev (U Michigan), Raum 1.013 (Johann von Neumann-Haus)
Title: Coble surfaces and Desargues configurations of lines and points

Abstract: A beautiful theorem of J. Thas asserts that the ten points of a Desargues configuration of lines and points formed by two perspective triangles in the projective plane can serve as the ten nodes of a unique rational plane curve of degree 6. I will give an algebraic-geometric proof of this theorem by using the geometry of a special Coble surfaces obtained as blow-ups of ten nodes of a rational sextic. We will also discuss an interesting dynamics of automorphisms of these special Coble surfaces.
07.12.2016 Bruno Klingler (U Paris, Jussieu)
Title: Hodge theory and atypical intersections

Abstract: Given a variation of Hodge structures V over a smooth quasi-projective base S, I will explain the notion of an atypical subvariety for (S, V) and state a simple general conjecture about these atypical subvarieties. When S is a Shimura variety and V a standard variation of Hodge structure on S, one recovers the Zilber-Pink conjectures, in particular the André-Oort conjecture. I will discuss results towards this conjecture.
14.12.2016 Olof Bergvall (HU Berlin)
Title: Moduli of plane quartics with level two structure

Abstract: The theory of plane quartics and their bitangents is a rich and fascinating subject with a history that goes back at least to the first half of the 19th century. From a more modern perspective, bitangents can be used to define so called "level structures" which have many interesting properties and applications. In this talk we shall study the moduli space of plane quartics with level two structure and in particular investigate its cohomology. On one hand, we shall use a description due to Looijenga in terms of arrangements of tori, and on the other we shall use a description of the moduli space in terms of points sets in the projective plane and we shall describe how to compute cohomology of these spaces via combinatorial methods and point counts over finite fields. The spaces we shall encounter have groups acting naturally on them and their cohomology groups are therefore representations of these groups. We will discuss these representations and we will also consider mixed Hodge structures.
25.01.2017 Zsolt Patakfalvi (EPF Lausanne)
Title: Projectivity of the moduli space of KSBA stable pairs and applications

Abstract: KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs are higher dimensional generalizations of (weighted) stable pointed curves. I will present a joint work with Sándor Kovács on proving the projectivity of this moduli space in characteristic zero, by showing that certain Hodge-type line bundles are ample on it. I will also mention applications to the subadditivity of logarithmic Kodaira dimension, and to the ampleness of the CM (Chow-Mumford) line bundle.
15:15 - 16:15
François Charles (U Paris, Orsay)
Title: Arithmetic ampleness and an arithmetic Bertini theorem

Abstract: We will discuss the analogue of the Bertini irreducibility theorem for ample hermitian line bundles on arithmetic varieties of absolute dimension at least 2. Along the way, we will explain some arithmetic analogues of well-known properties of positive line bundles in the arithmetic setting and describe how ideas and techniques stemming from the proof of the Bertini irreducibility theorem over finite fields (joint with Poonen) can apply.
15.02.2017 Seminar Einstein Stiftung Berlin: Rahul Pandharipande (ETH Zürich)
Title: New developments in the study of stable quotients

Abstract: I will start with a discussion of the moduli space of stable quotients and then explain its role in various recent developments: tautological relations, wall-crossing, and B-model geometry.



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