Oberseminar Algebra, Zahlentheorie und Diskrete Mathematik

Vorbesprechung

Loïs Faisant (IST Austria)

Counting rational curves with prescribed tangency conditions: a motivic analogue via universal torsors

Abstract: Given a smooth projective and geometrically irreducible curve C and a Mori Dream Space X, we present a general parametrisation of morphisms from C to X which allows us to express the Grothendieck motive of Hom (C,X) as a motivic function defined on some power of the scheme of effective divisors of C, generalising previous works of Bourqui. Such a parametrisation should be understood as lifting our morphisms to the universal torsor of X.
As an application, we prove a motivic analogue of a variant of Manin’s conjecture for Campana curves on smooth projective split toric varieties.

Average sizes of mixed character sums

Abstract: In joint work with Max Xu, we prove that the average size of a certain smoothly weighted, mixed character sum of length x is on the order of sqrt(x), under a weak Diophantine genericity condition on the angle θ of the additive character e(nθ). Certain quadratic Diophantine equations play a key role. In contrast, it was proved by Harper that the average size is o(sqrt(x)) for rational θ. Some remaining open questions will be highlighted.

Mo 19.5.2025 in Göttingen

15:00 Maxim Gerspach (Göttingen)

16:30 Gunther Cornelissen (Utrecht)