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

Online Forschungsseminar "Algebraische Geometrie"

Sommersemester 2020

Das Forschungsseminar findet mittwochs in der Zeit von 16:00 - 17:00 Uhr in Zoom statt.

Seminar: Algebraic Geometry an der FU

29.04.2020 Arnaud Beauville (Université de Nice)
Title: Vector bundles on Fano threefolds and K3 surfaces.

Abstract: Let \(X\) be a Fano threefold, and let \(S\) be a smooth anticanonical K3 surface in \(X\) . Any moduli space \(M\) of simple vector bundles on \(S\) carries a holomorphic symplectic structure. Following an idea of Tyurin, I will show that in some cases, those vector bundles which come from \(X\) form a Lagrangian subvariety of \(M\) . Most of the talk will be devoted to concrete examples of this situation.
06.05.2020 Daniel Erman (University of Wisconsin - Madison)
Title: Limits of polynomials rings.

Abstract: I will discuss two ways to think about a limit of a polynomial ring in \(n\) variables as \(n\) goes to infinity, and how this can be used to study complexity problems in algebraic geometry. This is joint work with Steven V Sam and Andrew Snowden.
13.05.2020 Riccardo Salvati Manni (Università degli studi di Roma "La Sapienza")
Title: The Kodaira dimension of the moduli space of abelian varieties of dimension 6.

Abstract: The moduli space \(\mathcal{A}_n\) of principally polarized complex abelian varieties of dimension \(n\) was believed for a long time to be unirational. This was first disproved by Freitag for \(n\) congruent to 1 mod 8 and n at least 17. Then Tai, Freitag and Mumford proved that the moduli space is of general type for all \(n\) at least 7. On the other hand, this moduli space is unirational for \(n\) at most 5. Recently M. Dittmann and N. R. Scheithauer and I proved that when \(n=6\) the Kodaira dimension is non-negative. I will give a general overview on the topic.
20.05.2020 Patrick Brosnan (University of Maryland)
Title: Fixed points in toroidal compactifications and essential dimension of covers.

Abstract: We (Najmuddin Fakhruddin and I) prove a fixed point theorem for the action of certain local monodromy groups on etale covers and use it to deduce lower bounds on essential dimension of covers. In particular, we can give geometric proofs of many (but definitely not all) of the results in a recent preprint of Farb, Kisin and Wolfson. Moreover, we can extend their incompressibility results for Hodge type Shimura varieties to certain Shimura varieties for E7 (those with 0-dimensional rational boundary components).
27.05.2020 Adrien Sauvaget (Université de Cergy Pontoise)
Title: Moduli spaces of large pluricanonical divisors.

Abstract: We consider the moduli spaces of curves endowed with a \(k\)-canonical divisor. The purpose of the talk is to explain that the geometry of moduli spaces of metric surfaces can be studied by considering the large \(k\)-behavior of these moduli spaces. The first application of this strategy is a proof of the finiteness of the volumes of moduli spaces of flat surfaces with conical singularities. The second one is a conjecture on the volumes of moduli spaces of hyperbolic cone surfaces.
03.06.2020 András Lőrincz (HU Berlin)
Title: Equivariant D-modules.

Abstract: In this talk, I will discuss some results concerning equivariant D-modules. Under suitable finiteness conditions, the category of such objects on a stratification is equivalent to the category of finite-dimensional representations of a quiver. We describe such quivers explicitly for some irreducible representations of complex reductive groups and toric varieties. In these cases we use the results to find the explicit D-module structure of local cohomology modules supported in orbit closures. Other applications include Lyubeznik numbers and intersection cohomology groups of orbit closures.
10.06.2020 17:00 - 18:00 Mihnea Popa (Northwestern University)
Title: Minimal exponents of singularities.

Abstract: The minimal exponent of a function is the negative of the largest root of its reduced Bernstein-Sato polynomial. It refines the notion of log canonical threshold, and it is related (sometimes conjecturally) to other interesting objects, for instance the Igusa zeta function. I will describe some results towards understanding minimal exponents, based on viewing them in the context of D-modules and Hodge theory on one hand, and birational geometry on the other. This is joint work with Mircea Mustata.
17.06.2019 16:00 - 17:00 Yuri Tschinkel (New York University)
Title: Equivariant birational geometry.

Abstract: I will discuss new invariants in equivariant birational geometry (joint work with Kontsevich and Pestun).
17:15 - 18:15 Mircea Mustaţă (University of Michigan)
Title: Minimal exponents of hypersurfaces and a conjecture of Teissier.

Abstract: The minimal exponent of a hypersurface is a refinement of the log canonical threshold, a fundamental invariant of singularities in birational geometry. After a brief introduction to these invariants, I will discuss work on a conjecture of Teissier, relating the invariant of a hypersurface with that of a hyperplane section. This is joint work (partially in progress) with Eva Elduque and Bradley Dirks.
24.06.2020 Egor Yasinsky (Universität Basel)
Title: Cremona groups and their subgroups. (Slides)

Abstract: The Cremona group is a group of birational automorphisms of a projective space. Its study goes back to classical works of Cremona, Noehter, Castelnuovo and Bertini, and it has been a subject of very intensive research during the last 15 years. This study employs different techniques - from modern birational geometry (e.g. Mori theory, Sarkisov program, and Caucher Birkar's works) to geometric group theory. In this talk, I will try to overview three plots on Cremona groups: classification of finite subgroups, topological properties, infinite subgroups and quotients of Cremona groups.
01.07.2020 15:00 Uhr Rahul Pandharipande (ETH Zürich)
Title: The moduli spaces of differentials on curves. (Notes)

Abstract: The moduli space of pairs \( (C,f) \) where \( C \) is a curve and \( f \) is a rational function leads to the well-developed theory of Hurwitz spaces. The study of the moduli of \( (C,\omega) \) where \( C \) is a curve and \( \omega\) is a meromorphic differential is a younger subject. I will discuss recent developments in the study of the moduli spaces of holomorphic/meromorphic differentials on curves. Many of the basic questions about cycle classes and integrals have now been solved (through the work of many people) -- but there are also several interesting open directions.
08.07.2020 Elisabetta Colombo (Università degli studi di Milano Statale)
Title: The dimension of the Voisin sets in the moduli space of abelian varieties.

Abstract: I will discuss reseach in collaboration with Naranjo and Pirola on the subsets \(V_k(A)\) of an abelian variety \(A\), consisting of those points \(x\) such that the zero-cycle \(x-O\) is \(k\)-nilpotent with respect to the Pontryaghin product. These sets were initially introduced and studied by Claire Voisin. We find a sharp upper bound of the dimension in moduli of the locus of those abelian varieties \(A\) for which \(V_2(A)\) is positive dimensional.
15.07.2020 17:00 Uhr Alexandru Suciu (Northeastern University Boston)
Title: Poincaré duality and cohomology jump loci. (Slides)

Abstract: The cohomology jump loci of a space \(X\) are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for \(X\). We will explore in this talk the geometry of these varieties and the delicate interplay between them in the context of compact, orientable 3-manifolds. In the process, we will arrive at a fairly precise geometric description of the resonance varieties \(R^i_k(A)\) of a 3-dimensional Poincaré duality algebra \(A\).


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