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

Forschungsseminar "Algebraische Geometrie"

Wintersemester 2018/19


Das Forschungsseminar findet mittwochs in der Zeit von 13:00 - 15:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 3.007, statt.

Seminar: Algebraic Geometry an der FU


17.10.2018 Daniele Agostini (HU Berlin)
Title: Algebraic statistics and abelian varieties

Abstract: Perhaps surprisingly, it turns out that many techniques and questions in algebraic geometry have natural applications in statistics. This is the subject of the relatively new field of Algebraic Statistics. In my talk, I will give an introduction to Algebraic Statistics for geometers trying to give an overview of some main ideas in the area. Then, I will present some recent results (joint with Carlos Amendola) that show how abelian varieties come up naturally when studying the Gaussian distribution on the integers.
 
24.10.2018 Ania Otwinowska (HU Berlin)
Title: On the Zariski closure of the Hodge locus

Abstract: Given a variation of Hodge structures V on a smooth quasi projective complex manifold S, the Hodge locus is the subset of points of S where exceptional Hodge tensors do occur. A famous result of Cattani, Deligne and Kaplan states that this Hodge locus is a countable union of algebraic subvarieties of S. In this talk we study the Zariski closure in S of the union of positive dimensional components of the Hodge locus. This is joint work with B. Klingler.
 
31.10.2018 Alessandro Verra (Uni Roma Tre)
Title: Coble cubics, genus 10 Fano threefolds and the theta map

Abstract: The talk deals with the relations between two different moduli spaces. From one side the branch divisor B is considered for the theta map of the moduli of semistable rank r vector bundles with trivial determinant on a genus 2 curve C. Special attention is payed to the case r = 3. Then B is the sextic dual to the Coble cubic, the unique cubic hypersurface singular along the Jacobian JC embedded by its 3-theta linear system. From the other side the moduli space of Fano threefolds X of genus 10 is considered. Since the intermediate Jacobian of X is JC, for a given genus 2 curve C, the assignement X ---> C defines a rational map f: F ---> M, M being the moduli space of genus 2 curves. Relying on a suitable description of the ramification divisor of the theta map, a description of f and of its fibres is outlined. The main result is that the fibres of f are naturally birational to the Coble cubic defined by JC. This is a joint work in progress with Daniele Faenzi.
 
07.11.2018 Sam Payne (University of Texas, Austin)
Title: Tropical methods for the Strong Maximal Rank Conjecture

Abstract: I will present joint work with Dave Jensen using tropical methods on a chain of loops to prove new cases of the Strong Maximal Rank Conjecture of Aprodu and Farkas. As time permits, I will also discuss relations to an analogous approach via limit linear series on chains of genus 1 curves.
 
14.11.2018 kein Seminar
21.11.2018 Frank-Olaf Schreyer (Universität des Saarlandes)
Title: Godeaux surfaces via homological algebra

Abstract: In this talk I report on joint work with my student Isabel Stenger about the construction of numerical Godeaux surfaces via homological algebra. Numerical Godeaux surfaces are minimal surfaces of general type with K2=1 and pg=q=0. So they are the surfaces of general type with smallest possible numerical invariants. It is conjectured that there are precisely 5 families of these surfaces, which are distinguished by their fundamental groups, which is conjectured to be Z/n, for n=1,...,5. Our homological algebra approach aims for a complete classification of these surfaces.
 
28.11.2018 kein Seminar
05.12.2018 Ana-Maria Castravet (Université de Versailles)
Title: Exceptional collections on moduli spaces of stable rational curves

Abstract: A question of Orlov is whether the derived category of the Grothendieck-Knudsen moduli space of stable, rational curves with n markings admits a full, strong, exceptional collection that is invariant under the action of the symmetric group S_n. I will present an approach towards answering this question. This is joint work with Jenia Tevelev.
 
12.12.2018 Ziyang Gao (IMJ-PRG Paris)
Title: Application of mixed Ax-Schanuel to bounding the number of rational points on curves

Abstract: With Philipp Habegger we recently proved a height inequality, using which one can bound the number of rational points on 1-parameter families of curves in terms of the genus, the degree of the number field and the Mordell-Weil rank (but no dependence on the Faltings height). In this talk I will give a blueprint to generalize this method to arbitrary curves. In particular I will focus on: (1) how establishing a criterion for the Betti map to be immersive leads to the desired bound; (2) how to apply mixed Ax-Schanuel to establish such a criterion. This is work in progress, partly joint with Vesselin Dimitrov and Philipp Habegger.
 
19.12.2018 Hanieh Keneshlou (MPI MiS Leipzig)
Title: Unirational components of moduli of genus 11 curves with several pencils of degree 6

Abstract: Considering a smooth d-gonal curve C of genus g, one may naturally ask about the existing possible number of pencils of degree d on C. Motivated by some questions of Michael Kemeny, in this talk we will focus on this question for hexagonal curves of genus 11. Inside the moduli space of genus 11 curves, we describe a unirational irreducible component of the locus of curves possessing k mutually independent and type I pencils of degree 6, for the values k=5,...,10.
 
09.01.2019 !RAUM 3.011!13:15 - 14:15 Frederic Campana (Université de Lorraine)
Title: Criterion for algebraicity of foliations, applications

Abstract (joint with M. Paun) A foliation F on X, complex projective smooth, is showed to have algebraic leaves if its dual is not pseudo-effective. In the particular case where F has positive minimal slope with respect to some movable class on X, the closures of the leaves are rationally connected. Combined with the existence of Viehweg-Zuo sheaves, this permits to show several versions of the Shafarevich-Viehweg 'hyperbolicity conjecture'.
 
14:45 - 15:45 Jochen Heinloth (Uni Essen)
Title: Existence of good moduli spaces for algebraic stacks

Abstract: Recently Alper, Hall and Rydh gave general criteria when a moduli problem can locally be described as a quotient and thereby clarified the local structure of algebraic stacks. We report on a joint project with Jarod Alper and Daniel Halpern-Leistner in which we use these results to show general existence and completeness results for good coarse moduli spaces. In the talk we will focus on two aspects that illustrate how the geometry of algebraic stacks gives a new point of view on classical methods for the construction of moduli spaces. Namely we explain how one-parameter subgroups in automorphism groups allow to formulate a version of Hilbert-Mumford stability in stacks that are not global quotients and sketch how one can reformulate Langton's proof of semistable reduction for coherent sheaves in geometric terms. This allows to apply the method to an interesting class of moduli problems.
 
16.01.2019 Michael Dettweiler (U Bayreuth)
Title: Tba.

Abstract: Tba.
 
23.01.201913:15 - 14:15 Rahul Pandharipande (ETH Zürich)
Title: Tba

Abstract: Tba.
 
14:45 - 15:45 Vadim Vologodski (HSE Moscow)
Title: Tba

Abstract: Tba.
 
30.01.2019 Philipp Habegger (Uni Basel)
Title: Tba.

Abstract: Tba.
 

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