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

Forschungsseminar "Algebraische Geometrie"

Wintersemester 2015/16

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

14.10.2015 kein Seminar
21.10.2015 Samuel Grushevsky (Stony Brook University)
Title: Compactified strata of meromorphic differentials

Abstract: We consider the locus of smooth curves together with a meromorphic one-form, with prescribed multiplicities of all its zeroes and poles, and study its closure over the Deligne-Mumford moduli space of stable curves. We describe this compactification completely, and illustrate our results with various examples. Based on joint work with M. Bainbridge, D. Chen, Q. Gendron, M. Moeller.
28.10.2015 Maksym Fedorchuk (Boston College)
Title: GIT semistability of Hilbert points of Milnor algebras

Abstract: The famous Mather-Yau theorem says that two isolated hypersurface singularities are biholomorphically equivalent if and only if their moduli algebras are isomorphic. However, the reconstruction problem of explicitly recovering the singularity from its moduli algebra is open. In the case of quasi-homogeneous hypersurface singularities, Eastwood and Isaev proposed an invariant-theoretic approach to the reconstruction problem. For homogeneous hypersurface singularities, Alper and Isaev gave a geometric invariant theory reformulation of this approach, in which GIT stability of the Hilbert points of the Milnor algebra of the singularity plays a key role. In particular, Alper and Isaev pose several problems concerning GIT stability of the associated form and the gradient point of a homogeneous form, which they recently solved in the binary case and in the case of generic forms in any number of variables. In my talk, I will explain these recent developments. I will then proceed to prove semistability of the gradient point of a semistable form, and semistability of the associated form of a non-degenerate form. This will answer several (but not all) questions of Alper and Isaev.
04.11.2015 Ananyo Dan (HU Berlin)
Title: New methods on non-reduced components of Hilbert schemes of curves

Abstract: We study non-reduced components of Hilbert schemes of local complete intersection curves (not necessarily reduced) using techniques from Hodge theory.
11.11.2015 Jérémy Guéré (HU Berlin)
Title: From Koszul cohomology to tautological relations in the moduli space of curves

Abstract: I will explain how to derive tautological relations in the moduli space of stable curves from a vanishing of the cohomology of some Koszul complex. The vanishing property comes from a result of Green on the study of base-point free linear systems and the Koszul complex comes from the algebraic definition of Witten r-spin class. Precisely, our tautological relations hold in the Chow ring of the moduli space of r-spin curves and do not rely on any semi-simplicity condition. I will also discuss the information we obtain on this Witten r-spin class, its generalization to some non-semi-simple cohomological field theories, and its application to the double ramification hierarchy recently introduced by Buryak.
18.11.2015 Stefan Schreieder (Universität Bonn)
Title: A very general quartic or quintic fivefold is not stably rational

Abstract: Totaro showed recently that a wide range of hypersurfaces are not stably rational. In this talk I explain how to improve Totaro's result in dimension 5, where it covers very general hypersurfaces of degree at least 6. Namely, we prove that a very general quartic or quintic fivefold is not stably rational. It was not known whether these varieties were rational. Moreover, general quartic fivefolds are known to be unirational; they are in fact the first smooth hypersurfaces that are known to be unirational but not stably rational. This is joint work with L. Tasin.
25.11.2015 Sándor Kovács (University of Washington)
Title: Positivity results for push-forwards of relative pluri-canonical sheaves with applications

Abstract: This is a report on joint work with Zsolt Patakfalvi. We prove various positivity results for push-forwards of relative pluri-canonical sheaves of families of stable log-varieties. The primary motivation for these results is that they are used to prove projectivity of the moduli space of stable log-varieties and to confirm the Iitaka-Viehweg conjecture on the subadditivity of log-Kodaira dimension for fiber spaces whose general fiber is of log general type. However they turn out to be useful in other situations as well. I will mention briefly other applications of these positivity results by Patakfalvi and Xu to prove that the CM line bundle is ample on the proper moduli space parametrizing KSBA stable varieties and by Ascher and Turchet to prove a logarithmic version of the Correlation Theorem of Caporaso-Harris-Mazur (dim 1), Hassett (dim 2), and Abramovich (general).
26.11.2015 Sándor Kovács (University of Washington) *Room 1.012 Time: 16:00-17:00*
Title: What's the "right" definition of a moduli functor of higher dimensional stable varieties?

Abstract: As a first approximation one might expect that the definition of a moduli functor in arbitrary dimensions is simply the same as the one for curves after replacing the members of families with higher dimensional ones. In this talk I will explain some obstacles that show that this is not the case and possible ways to deal with these issues. The plan is to discuss some explicit examples.
02.12.2015 kein Seminar
09.12.2015 Valentin Tonita (HU Berlin)
Title: Quantum K-theory

Abstract: For a project manifold X, let X0,n,d be the moduli spaces of (genus 0, degree d) stable maps to X. I will define both "ordinary" and permutation equivariant K-theoretic Gromov-Witten invariants (recently introduced by A. Givental) as holomorphic Euler characteristics on X0,n,d, respectively X0,n,d/Sn . I will characterize their generating series in terms of the cohomological Gromov-Witten theory of X, which is easier to compute. Time permitting I will explain recent progress in the permutation-equivariant theory: use of localization for K -theoretic I functions of toric manifolds, twisted K-theoretic Gromov-Witten invariants etc.
5.01.2016 Samuel Grushevsky (Stony Brook University) SFB Seminar. *Place: IRIS-Haus. Time: 15:30-16:30*
Title:Moduli of abelian varieties: homology and compactifications

5.01.2016 Gavril Farkas (Humboldt Universität zu Berlin) SFB Seminar. *Place: IRIS-Haus. Time: 17:00-18:00*
Title:Moduli of abelian varieties: uniformization

06.01.2016 Aaron Bertram (University of Utah)
Title:On stange duality for del Pezzo surfaces

Abstract: LePotier's strange duality conjectures a perfect pairing between the global sections of determinant line bundles on moduli spaces of coherent sheaves with orthogonal invariants on a del Pezzo surface. Following work of Belkale and Marian-Oprea proving an analogous theorem for moduli spaces of vector bundles on curves, we conjecture that a finite Grothendick quot scheme reveals dual bases for the sections of the respective bundles. We obtain positive results in this direction when one of the moduli spaces is a Hilbert scheme via recently computed multiple point formulas. This is joint work with Thomas Goller and Drew Johnson.
13.01.2016 Pawel Borówka (Jagiellonian University, Krakow )
Title:Non-simple abelian varieties and (1,3) theta divisors.

Abstract: I will present the construction that shows that on a general (1,3) polarised abelian variety, there exists a unique (up to translation) smooth hyperelliptic curve defined as the zero set of an odd theta function. By construction, the Jacobian of the curve is non-simple. This leads to a problem of understanding the locus of non-simple principally polarised abelian varieties. I will show that the irreducible components of the locus of non-simple varieties are these with fixed dimension of abelian subvariety and type of restricted polarisation I will also give equations on the Siegel space that produce non-simple varieties. Finally, I will conclude with some unanswered questions and further directions.
20.01.2016 Herbert Lange (Erlangen)
Title:A criterion for an abelian variety to be non-simple

Abstract: We explain a criterion in terms of period matrices for an arbitrary polarized abelian variety to be non-simple. We will show some examples. This is a joint work with R. Auffarth and A. Rojas.
27.01.2016 Maryna Viazovska (Humboldt Universität zu Berlin) *TALK POSTPONED*
Title:Strongly regular graphs and Prym-Tyurin varieties.

Abstract: A strongly regular graph with parameters (v,k,l,m) is a k-regular graph in which every pair of adjacent vertices has l common neighbors and every pair of non-adjacent vertices has m common neighbors. Surprisingly enough, these combinatorial objects appear in different branches of mathematics. For example, certain types of strongly regular graphs can be used to construct Prym-Tyurin varieties. In this talk we will give an overview of the theory of these graphs, explain the construction of associated Prym varieties, and also will report on new non-existence results for strongly regular graphs.
11.02.2016 Iman Setayesh. *Room: TBD. Time: 16:45-17:45.*
Title:The product structure of the kappa ring of the moduli of curves of compact type

Abstract: In this talk, I will explain a product rule in the kappa ring of the moduli space of curves of compact type.


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