Humboldt-Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik
Wintersemester 2024-2025
Time: Wednesday 13:15 - 14:45
Room: 3.007 John von Neumann-Haus
Humboldt Arithmetic Geometry Seminar
Seminar: Algebraic Geometry an der FU
| Time | Room | Speaker |
|---|---|---|
| 16.10.2024, 13:15h | Irene Spelta (HU Berlin) | |
| Title: Decomposable G-curves and special subvarieties of the Torelli locus | ||
| Abstract: As it is well known, the Torelli morphism \(j:\mathcal{M}_g\to\mathcal{A}_g\) sends an algebraic curve \(C\) to its Jacobian variety \(JC\). The (closure of the) image inside \(\mathcal{A}_g\) is the so-called Torelli locus. In this talk, we will discuss on the extrinsic geometry of this locus. In particular, we will consider certain special subvarieties coming from families of decomposable G-curves. | ||
| 23.10.2024, 13:15 | Pedro Montero (Universidad Técnica Valparaíso) | |
| Title: Additive structures on quintic del Pezzo varieties | ||
| Abstract: A classical problem of F. Hirzebruch concerns the classification of compactifications of affine space into smooth projective varieties with Picard rank one. It turns out that any such compactification must be a Fano manifold, i.e., it has an ample anti-canonical divisor. After reviewing some known results, I will focus on the specific case of equivariant compactifications of affine space (i.e., of the "vector group" \(\mathcal{G}_a^n\)), particularly in the case of del Pezzo varieties. We will recall that del Pezzo varieties are a natural higher-dimensional generalization of classical del Pezzo surfaces. Over the field of complex numbers, these varieties were extensively studied by T. Fujita in the 1980s, who classified them by their degree. I will present a result on the existence and uniqueness of "additive structures" on del Pezzo quintic varieties. Specifically, we determine when and how many distinct ways they can be obtained as equivariant compactifications of the commutative unipotent group. As an application, we obtain results on the k-forms of quintic del Pezzo varieties over an arbitrary field k of characteristic zero, as well as for singular quintic varieties. This is a joint work with Adrien Dubouloz and Takashi Kishimoto. | ||
| 06.11.2024, 13:15h | Andrei Yafaev (University College London) | |
| Title: On the Andre-Pink-Zannier conjecture and its generalisations | ||
| Abstract: This is a joint work with Rodolphe Richard (Manchester). The Andre-Pink-Zannier conjecture is a case of Zilber-Pink conjecture on unlikely intersections in Shimura varieties. We will present this conjecture and a strategy for proving it as well as its proof for Shimura varieties of abelian type. In the second talk we present a "hybrid conjecture" combining the recently proved Andre-Oort conjecture and Andre-Pink-Zannier. It is motivated by its analogy with Mordell-Lang for abelian varieties. We will explain this analogy as well as the proof of the hybrid conjecture for Shimura varieties of abelian type. (This is the second talk, the first one taking place on Tuesday at the Arithmetic Geometry Seminar). | ||
| 13.11.2024, 13:15h | Alessandro Verra (Università Roma Tre) | |
| Title: An application of the Segre primal to an enumerative problem | ||
| Abstract: The classification of complex, nodal cubic threefolds goes back to Corrado Segre. Among these, a unique one, up to projective equivalence, has the maximal number of ten nodes and it is named the Segre primal. In this talk we describe the solution of the following enumerative problem, where the Segre primal appears. Let \(V\) be a smooth complex cubic threefold and \(x\) a general point of it, then the six lines of \(V\) through \(x\) are in a quadric cone surface and define six points of the projective line \(P\). This defines a rational map from \(V\) to the moduli space of genus 2 curves. What is the degree of this map? Joint work with Ciro Ciliberto. | ||
| 15.11.2024 | A day of Arithmetic Geometry on the ocassion of the retirement of J. Kramer | |
| A day of Arithmetic Geometry | ||
| 15.01.2025, 13:15h | Andreas Kretschmer (HU Berlin) | |
| Title: Symmetric Ideals and Invariant Hilbert Schemes | ||
| Abstract: A symmetric ideal is an ideal in a polynomial ring which is stable under all permutations of the variables. Special classes of symmetric ideals (e.g. Specht ideals, Tanisaki ideals) feature prominently in algebraic combinatorics: For example, the non-negativity of the q-Kostka polynomials is eventually explained by its interpretation as the Hilbert series of the quotient of a Tanisaki ideal. In commutative algebra, chains of symmetric ideals in increasingly many variables stabilize "up to symmetry" and several of their invariants display a uniform behavior. This is closely related to the concepts of representation stability and FI-modules. This project grew out of a desire to study symmetric ideals more systematically. In particular, we wanted to get a feeling for the naive question of "how many" (zero-dimensional) symmetric ideals there actually are, after fixing certain parameters. This leads to the study of the invariant Hilbert scheme for the action of the symmetric group on affine n-space. We prove irreducibility and smoothness results and compute dimensions in certain special cases. Moreover, we classify all homogeneous symmetric ideals of colength at most 2n and decide which of these define singular points. We also discuss a conjectural stabilization phenomenon and explain connections to the usual Hilbert schemes of points of affine space. This is joint work with Sebastian Debus. | ||