Many rational points on del Pezzo surfaces of low degree
Freitag, 17.10.25, 10:30-11:30, Seminarraum 404
Let X be an algebraic variety over a number field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. Questions one might ask are, is X(k) empty or not? And if it is not empty, how ‘large’ is X(k)? Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d at least 3, these are the smooth surfaces of degree d in P^d). The lower the degree, the more complex del Pezzo surfaces are. I will give an overview of different notions of ‘many’ rational points, and go over several results for rational points on del Pezzo surfaces of degree 1 and 2. I will then focus on work in progress joint with Julian Demeio and Sam Streeter on the so-called Hilbert property for del Pezzo surfaces of degree 1.
Singularities of base spaces of Lagrangian fibrations
Freitag, 14.11.25, 10:30-12:00, Seminarraum 404
Irreducible holomorphic symplectic varieties (or IHS for short) are a special class of projective algebraic varieties that can be studied from various angles; they are interesting because they are expected to satisfy many special geometric properties. Yet, at this time, they remain largely illusive. One promising way to understand the geometry of IHS varieties is through the study of so-called Lagrangian fibrations. A folklore conjecture attributed to Matsushita claims that the base \(X\) of such a fibration is necessarily isomorphic to the complex projective space. In this talk, we will survey several aspects of the geometry of IHS varieties. Finally, we present a new and short proof of Matsushita's conjecture in case \(\dim X = 2\). This talk is based on joint work with Zheng Xu.
Model theory, differential algebra and functional transcendence
Freitag, 21.11.25, 10:30-12:00, Seminarraum 404
A fundamental problem in the study of algebraic differential equations is determining the possible algebraic relations among different solutions of a given differential equation. Freitag, Jaoui, and Moosa have isolated an essential property, called property D2, in order to show that if a differential equation given by an irreducible differential polynomial of order n is defined over the constants and has property D2, then any number of pairwise distinct solutions together with their derivatives up to order n-1 are algebraically independent. The property D2 requires that, given two distinct solutions, there is no non-trivial algebraic dependence between the solutions and their first n-1 derivatives.
The proof of Freitag, Jaoui and Moosa is extremely elegant and short, yet it uses in a clever way fundamental results of the model theory of differentially closed fields of characteristic 0. The goal of this talk is to introduce the model-theoretic tools at the core of their proof, without assuming a deep knowledge in (geometric) model theory (but some familiarity with basic notions in algebraic geometry).
A Unified Finiteness Theorem For Curves
Freitag, 21.11.25, 14:00-15:30, Seminarraum 404
This talk presents a unified framework for finiteness results concerning arithmetic points on algebraic curves, exploring the analogy between number fields and function fields. The number field setting, joint work with F. Janbazi, generalizes and extends classical results of Birch–Merriman, Siegel, and Faltings. We prove that the set of Galois-conjugate points on a smooth projective curve with good reduction outside a fixed finite set of places is finite, when considered up to the action of the automorphism group of a proper integral model. Motivated by this, we consider the function field analogue, involving a smooth and proper family of curves over an affine curve defined over a finite field. In this setting, we show that for a fixed degree, there are only finitely many étale relative divisors over the base, up to the action of the family's automorphism group (and including the Frobenius in the isotrivial case). Together, these results illustrate both the parallels and distinctions between the two arithmetic settings, contributing to a broader unifying perspective on finiteness.
The speaker will join us online. The zoom-link will be sent to the algebra mailing list. Otherwise available on request.