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

Forschungsseminar "Algebraische Geometrie"

Sommersemester 2023

Time: Wednesday 13:15 - 14:45

Room: 3.007 John von Neumann-Haus

Humboldt Arithmetic Geometry Seminar

Seminar: Algebraic Geometry an der FU

26.04.2023, 13:15 Ignacio Barros (University of Antwerp)
Title: Moduli spaces of hyperkähler varieties
Abstract: I will discuss general aspects of the geometry of moduli spaces of hyperkähler varieties and talk about the state of the art regarding their birational classification. In the second half, I will present recent results on their Kodaira dimension generalizing to higher dimension results of Gritsenko-Hulek-Sankaran in the surface and fourfold K3-type cases. The talk is based on joint work with Pietro Beri, Emma Brakkee, and Laure Flapan.
26.04.2023, 14:30 Samouil Molcho (ETH Zürich)
Title: Extending Brill-Noether classes to the boundary of moduli space of curves
Abstract: Inside the Jacobian of the universal curve of the moduli space of genus \(g\), \(n\)-pointed curves lie the Brill-Noether loci, parametrizing pairs of curves with line bundles that have more than expected sections. Pulling the (virtual fundamental classes of the) Brill-Noether loci to \(M_{g,n}\) by any section of the universal Jacobian produces interesting cycles in the tautological ring, which play a key role in its structure as a ring. Pagani, Ricolfi and van Zelm have proposed an extension of these classes to the Deligne-Mumford compactification \(\bar{M}_{g,n}\), and conjecture that they are also in the tautological ring. In this talk, I will explain a natural refinement of these classes from the perspective of logarithmic geometry, which allows us to prove the PRvZ conjecture.
03.05.2023 Marian Aprodu (University of Bucharest)
Title: Resonance and vector bundles
Abstract: Resonance varieties are algebro-geometric objects that emerged from the geometric group theory. They are naturally associated to vector subspaces in second exterior powers and carry natural scheme structures that can be non-reduced. In algebraic geometry, they made an unexpected appearance in connection with syzygies of canonical curves. In this talk, based on a joint work with G. Farkas, C. Raicu and A. Suciu, I report on some recent results concerning the geometry of resonance schemes in the vector bundle case.
10.05.2023 Nicola Tarasca (Virginia Commonwealth University)
Title: Coinvariants of vertex algebras and abelian varieties
Abstract: Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on my recent preprint arXiv:2301.13227.
11.05.2023 - 12.05.2023
The Berlin-Hannover Algebraic Geometry Workshop
14.06.2023, 13:15 Cécile Gachet (HU Berlin)
Title: Smooth projective surfaces with infinitely many real forms
Abstract: A common undergraduate exercise is to classify quadratic forms over the real and complex numbers. Its conclusion could be that the two non-isomorphic real conics \(x^2 + y^2 + z^2 = 0\) and \(x^2 + y^2 - z^2 = 0\) are isomorphic as complex curves. In fact, the corresponding complex curve is the rational line, and it admits only the afore-mentioned two non-isomorphic real forms. Although it is quite common to find complex projective varieties admitting several real forms, the first example of a variety with infinitely many non-isomorphic real forms can be found in a 2018 paper by Lesieutre. More examples of varieties with infinitely many real forms have been found later, for instance as rational surfaces and as surfaces birational to K3 surfaces, see the 2022 paper by Dinh, Oguiso, and Yu.
This talk, reporting on joint work with Tien-Cuong Dinh, Hsueh-Yung Lin, Keiji Oguiso, Long Wang, and Xun Yu, completes the picture sketched by theses examples in the case of smooth projective surfaces. It features the following two results: First, if a smooth projective surface admits infinitely many real forms, then it is rational, or birational to a K3 surface and non-minimal, or birational to an Enriques surface and non-minimal. Second, there are surfaces obtained by blowing-up one point in an Enriques surface, which admit infinitely many non-isomorphic real forms. In this talk, I will explain the key ideas involved in the proofs of the two results, and try to give an idea of the construction used for the second one. Interestingly, we will encounter a fair share of group theory, group actions, and dynamics.
14.06.2023, 14:30 Phil Engel (University of Georgia)
Title: The non-abelian Hodge locus
Abstract: Given a family of smooth projective varieties, one can consider the relative de Rham space, of flat vector bundles of rank \(n\) on the fibers. The flat vector bundles which underlie a polarized \(\mathbb{Z}\)-variation of Hodge structure form the "non-abelian Hodge locus". Simpson proved that this locus is analytic, and he conjectured it is algebraic. This would imply a conjecture of Deligne that only finitely many representations of the fundamental group of a fiber appear. I will discuss a proof of Deligne's and Simpson's conjectures, under the additional hypothesis that the \(\mathbb{Z}\)-zariski closure of monodromy is a cocompact arithmetic group. This is joint work with Salim Tayou.
21.06.2023 Michael McBreen (The Chinese University of Hong Kong)
Title: Introduction to microlocal sheaves (Lecture 2 of the minicourse)
Abstract: Given a manifold \(M\) and an open subset \(U\) of the cotangent bundle \(T^*M\), we define the category \(\mu Sh(U)\) of microlocal sheaves on \(U\), as well as the category \(\mu Sh_Z(U)\) of microlocal sheaves supported on any given subset \(Z\) of \(U\). We sketch the basic features of this category, and describe it as explicitly as possible using the constructions from Lecture 1.
28.06.2023 Eric Larson (Brown University)
Title: Interpolation for Brill-Noether Curves
Abstract: In this talk, we determine when there is a Brill-Noether curve of given degree and given genus that passes through a given number of general points in any projective space.
05.07.2023, 13:15 Yuri Tschinkel (New York University)
Title: Birational types
Abstract: I will discuss joint work with Chambert-Loir, Kontsevich, and Kresch on new invariants in birational geometry.
05.07.2023, 14:30 Rahul Pandharipande (ETH Zürich)
Title: Beyond the tautological ring of the moduli of curves

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