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

Forschungsseminar "Algebraische Geometrie"

Wintersemester 2017/18

Das Forschungsseminar findet mittwochs in der Zeit von 13:00 - 15:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 1.114, statt.

Seminar: Algebraic Geometry an der FU

18.10.2017 Frank Gounelas (HU Berlin)
Title: Measures of irrationality

Abstract: In this talk I will begin by introducing and discussing basic properties of various invariants associated to an arbitrary projective variety, that aim to measure how far it is from being rational, rationally connected or uniruled respectively. In the second part of the talk I will discuss recent work with Alexis Kouvidakis computing these invariants for the Fano scheme of lines of a cubic threefold, which is a general type surface classically studied by Fano, Clemens-Griffiths and others.
08.11.2017 Ariyan Javanpeykar (Uni Mainz)
Title: Hyperbolicity of moduli spaces

Abstract: The moduli space of curves of genus at least two is hyperbolic (in any reasonable sense of the word hyperbolic). I will explain what this means precisely, and what one could expect in higher dimensions.
15.11.2017 kein Seminar wegen der NoGAGS 2017 (s. u.)
16. - 17.11.2017 NoGAGS 2017 - North German Algebraic Geometry Seminar 2017
22.11.2017 Michael Gröchenig (FU Berlin)
Title: Rigid local systems

Abstract: An irreducible representation of an abstract group is called rigid, if it gives rise to an isolated point in the moduli space of all representations. Complex rigid representations are always defined over a number field. According to a conjecture by Simpson, for fundamental groups of smooth projective varieties one should expect furthermore integrality. I will report on joint work with H. Esnault where we prove this for so-called cohomologically rigid representations. Our argument is mostly arithmetic and passes through fields of positive characteristic and the p-adic numbers.
29.11.2017 Marc Levine (Uni Duisburg-Essen)
Title: A calculus of characteristic classes in Witt cohomology

Abstract: As part of a program to develop an enumerative geometry with values in quadratic forms, we would like to have a practical calculus for suitable characteristic classes of vector bundles. One promising setting is via the theory of Pontryagin/Euler classes the cohomology of the sheaf of Witt groups. We describe the main points of this calculus: the SL_2-splitting principle of Ananyevskiy, an extension to a splliting principle for reduction to the normaliser of the torus in SL_2, and the computation of Pontryagin/Euler classes of symmetric powers of SL_2-bundles. As an application, we compute the quadratic form ``counting’’ the lines on a smooth hypersurface of degree 2d-1 in P^{d+1}.
13.12.2017 Angela Ortega (HU Berlin)
Title: Generic inyectivity of the Prym map for double ramified coverings

Abstract: Given a finite morphism of smooth curves one can canonically associate it a polarized abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarized abelian varieties, known as the Prym map. In this talk we will consider the Prym map between the moduli space of double coverings over a genus g curve ramified at r points, and the moduli space of polarized abelian varieties of dimension (g-1+r)/2 with polarization of type D. We will show the generic injectivity of the Prym map in the cases (a) g=2, r=6 and (b) g=5, r=2. In the first case the proof is constructive and can be extended to the range r > max{6, 2(g+2)/3}. This a joint work with J. C. Naranjo.
10.01.2018 Alexandru Constantinescu (Uni Genova)
Title: Linear Syzygies and Hyperbolic Coxeter Groups

Abstract: We show that the virtual cohomological dimension of a Coxeter group is essentially the same as the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of its nerve. Using this connection, we modify a construction of Osajda in group theory to find for every positive integer r a quadratic monomial ideal, with linear syzygies, and regularity of the quotient equal to r. This answers a question of Dao, Huneke and Schweig, and shows that Gromov asked essentially the same question about the virtual cohomological dimension of hyperbolic Coxeter groups. For monomial quadratic Gorenstein ideals with linear syzygies we prove that the regularity of their quotients can not exceed four, which implies that for d > 4 every triangulation of a d-manifold has an induced square or a hollow simplex. All results are in collaboration with Thomas Kahle and Matteo Varbaro.
17.01.2018 13:00 - 14:00 Filippo Viviani (Uni Roma Tre)
Title: Effective cycles on the symmetric product of a curve: the Abel-Jacobi faces

Abstract: In the above talks we study the convex-geometric properties of the cone of (pseudo)-effective n-cycles in the symmetric product of a curve. In the first talk, we introduce and study the Abel-Jacobi faces, related to the contractibility properties of the Abel-JAcobi morphism and to classical Brill-Noether varieties. In the second talk, we study the cone generated by the n-dimensional diagonal cycles and we prove that it is a perfect face of the cone of (pseudo)-effective cycles. This is joint work with F. Bastianelli and A.F. Lopez.
14:30 - 15:30 Alexis Kouvidakis (Uni of Crete)
Title: Effective cycles on the symmetric product of a curve: the diagonal cone

Abstract: Same as Vivianis abstract (see above).
24.01.2018 Soheyla Feyzbakhsh (University of Edinburgh)
Title: Applications of wall-crossing in classical algebraic geometry

Abstract: Mukai has introduced a geometric program to reconstruct a K3 surface from a curve on that surface. The idea is to first consider a Brill-Noether locus of vector bundles on the curve. Then the K3 surface containing the curve can be obtained uniquely as a Fourier-Mukai partner of the Brill-Noether locus. I will explain how wall-crossing with respect to Bridgeland stability conditions implies that the Mukai's strategy works for curves of genus greater than 12 or genus 11. If time permits, I will also talk about Clifford indices of curves.
31.01.2018 Margherita Lelli-Chiesa (Universita de L'Aquila)
Title: Nikulin surfaces and moduli of Prym curves

Abstract: The relevance of K3 surface in the study of the moduli space of curves is well-established. Nikulin surfaces, that is, K3 surfaces endowed with a nontrivial double cover branched along eight disjoint rational curves, play a similar role at the level of the moduli space of Prym curves. I will report on a work in this direction joint with Knutsen and Verra. In particular, I will prove that a general Nikulin section of fixed genus lies exactly on one Nikulin surface with only a few exceptions occuring in low genus.
01.02.2018 (DONNERSTAG) Einstein Lectures in Algebraic Geometry: Rahul Pandharipande (ETH Zürich), Johannes Schmitt (ETH Zürich) und Jason van Zelm (Liverpool/HU Berlin)
Ort: Raum 021, IRIS-Haus, Zum Großen Windkanal 6, 12489 Berlin
11:00 - 12:15, Rahul Pandharipande (ETH Zürich): Hurwitz loci, integrals, and open questions

Abstract: Hodge integrals over the moduli of Hurwitz covers have many beautiful properties. I will discuss their relationship to the Hilbert scheme of points of the plane. There are many interesting computations to consider --- most require the development of better tools. I will also discuss to the role of Hurwitz loci in the study of tautological classes as an introduction to the lectures by J. Schmitt and J. van Zelm.
14:00 - 15:15, Jason van Zelm (Liverpool/HU Berlin): Nontautological bielliptic classes

Abstract: Tautological classes are geometrically defined classes in the Chow ring of the moduli space of curves which are particularly well understood. The classes of many known geometrically defined loci have been shown to be tautological. A bielliptic curve is a curve with a 2-to-1 map to an elliptic curve. In this talk we will introduce tautological and nontautological classes. We will then build on an idea of Graber and Pandharipande to show that the closure of the locus of bielliptic curves in the moduli space of stable curves of genus g is non-tautological when g is at least 12.
15:45 - 17:00, Johannes Schmitt (ETH Zürich): Extending the tautological ring by Hurwitz cycles

Abstract: The tautological ring is the subring of the cohomology of the moduli spaces of stable curves formed by tautological classes. Many of its features - like a generating set, intersection products and pullbacks/pushforwards under natural maps - are accessible in terms of combinatorics. Unfortunately, as shown in the previous talk, the tautological ring is in general a strict subring of the (algebraic) cohomology. We show how this ring can be extended by adding the classes of Hurwitz loci, i.e. loci of curves forming ramified covers of curves of lower genus (like the sets of hyperelliptic or the bielliptic curves discussed before). We demonstrate how this extended ring still admits a combinatorial description and how this can be used to compute many classes of Hurwitz loci that already lie in the tautological ring.
07.02.2018 Xavier Roulleau (University of Aix-Marseille)
Title: Construction of Nikulin configurations on some Kummer surfaces

Abstract: Joint work with Alessandra Sarti. A Nikulin configuration on a K3 surface is a set C of 16 smooth disjoint rational curves. A famous result of Nikulin is that any K3 surface X containing a Nikulin configuration is a Kummer surface, which means that there exists an abelian surface A such that X is the minimal resolution of the quotient A/[-1] and the exceptional curves of the resolution X->A[-1] are the 16 curves of the Nikulin configuration C (this is denoted X=Km(A)). In this talk, starting with a Kummer configuration C on some polarised Kummer surface X, we will construct another Kummer configuration C' on X such that if A and A' denotes the associated Abelian surfaces, although one has: Km(A)=X=Km(A'), the Abelian surfaces A and A' are not isomorphic (unless X is a Jacobian Kummer surface). That construction uses the Torelli Theorem for K3. It gives some new knowledge on the automorphisms group of Kummer surfaces and it brings some new special configurations of rational curves on K3's. If time permits, we will derive some applications to the construction of interesting surfaces of general type, like the Schoen surfaces.
14.02.201813:15 - 14:15 Andrey Soldatenkov (Uni Bonn)
Title: Kuga-Satake construction under degeneration

Abstract: Degree two cohomology H of any projective K3 surface carries a polarized Hodge structure with one-dimensional (2,0)-component. The Kuga-Satake construction attaches to it an abelian variety A and an embedding of H into the second cohomology of A, compatible with Hodge structures. I will talk about our joint work with S.Schreieder, where we study the behaviour of Kuga-Satake abelian varieties for degenerating families of K3 surfaces.
14:45 - 15:45 Francesco Russo (Uni Catania)
Title: Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds

Abstract: Kuznetsov Conjecture and the work of Hassett predict that a general cubic fourfold belonging to an irreducible divisor C_d parametrizing smooth cubic fourfolds of discriminant d is rational if and only if d is an admissible value in the sense of Hassett, that is, if and only if d>6 is an even integer not divisible by 4, by 9 nor by any odd prime of the form 2+3m. I will present a proof of this conjecture for the smallest admissible values d=26 and d=38 (the case d=14 being classical), via the construction of a congruence of 5-secant conics to a surface contained in the general element of C_d for the respective values of d. This is joint work with Giovanni Staglianò.


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