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 Sn. 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: Elliptic curve convolution and sheaves with Tannaka group G2

Abstract: The talk is about a joint work with Collas, Reiter and Sawin. We study the convolution of a certain subcategory of perverse sheaves on elliptic curves. By the Tannaka formalism, to each such sheaf one has attached an algebraic group: the Tannaka group of the sheaf. Using explicit monodromy calculations, we prove a general theorem under which circumstances sheaves of rank two with 7 singularities have a Tannaka group equal to the simple algebraic group G2.
23.01.2019 !RAUM 3.011!13:15 - 14:15 Rahul Pandharipande (ETH Zürich)
Title: Double ramification cycles for target varieties

Abstract: A basic question in the theory of algebraic curves is whether a divisor represents the zeros and poles of a rational function. An explicit solution in terms of periods was given by the work of Abel and Jacobi in the 19th century. In the past few years, a different approach to the question has been pursued: what is the class in the moduli of pointed curves of the locus of such divisors? The answer in Gromov-Witten theory is given by Pixton's formula for the double ramification cycle. I will discuss recent work with F. Janda, A. Pixton, and D. Zvonkine which considers double ramification cycles for target varieties X (where Pixton's original question is viewed as the X=point case). I will also discuss the associated relations studied by Y. Bae.
14:45 - 15:45 Vadim Vologodsky (HSE Moscow)
Title: On the periodic topological cyclic homology of (DG) algebras in characteristic p

Abstract: We prove that the periodic topological cyclic homology of a smooth proper DG algebra over Fp is isomorphic to the periodic (algebraic) cyclic homology of a lifting of the algebra over Zp. This is a joint work with Alexander Petrov.
24.01.2019 !RAUM 3.006!15:00 - 17:00 Carel Faber (Utrecht University)
Title: Ternary quartics and modular forms of genus three

Abstract: I will discuss the precise relation between concomitants for plane quartics and (Teichmüller and Siegel) modular forms of genus three.
30.01.2019 Philipp Habegger (Uni Basel)
Title: Equidistribution of roots of unity and the Mahler measure

Abstract: It is a classical fact that roots of unity of given order n equidistribute around the unit circle as n\rightarrow\infty. In other words, the average of a continuous, real-valued function f on C taken over roots of unity of order n converges to the normalized integral of f over the unit circle as n\rightarrow\infty. Baker, Ih, and Rumely extended this to the test functions \log|P(\cdot)| where P is a univariate polynomial with algebraic coefficients. I will discuss work in progress with Vesselin Dimitrov where we allow a certain class of multivariate polynomials. In particular, f may have logarithmic singularities along a divisor. The resulting integral is the Mahler measure of P. If time permits I will also discuss connections to work of Lind, Schmidt, and Verbitskiy on dynamical systems of algebraic origin.
06.02.2019 kein Seminar
13.02.2019 !RAUM 3.011!13:15 - 14:15 Lynn Chua (UC Berkeley)
Title: Schottky Algorithms: Classical meets Tropical

Abstract: We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically. The locus of cographic matroids arises from tropicalizing the Schottky-Igusa modular form. This is joint work with Mario Kummer and Bernd Sturmfels.
14:30 - 15:30 Francesco Galuppi (MPI Leipzig)
Title: The rough path signature variety

Abstract: Signature tensors are a useful tool in the study of paths X. When X runs among a given class of path (e.g. polynomial, piecewise linear, etc), the signature of X parametrizes an algebraic variety. The geometry of this variety reflects some of the properties of the chosen class of path. The most important example is the class of rough paths, that are widely studied in stochastic analysis. Their signature variety presents many similarities to the Veronese variety, and we'll illustrate the first nice results, as well as some open questions.


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