Motivic distribution of rational curves and twisted products of toric varieties

Faisant, Loïs

doi:10.2140/ant.2025.19.883

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Motivic distribution of rational curves and twisted products of toric varieties

Loïs Faisant

Vol. 19 (2025), No. 5, 883–965

DOI: 10.2140/ant.2025.19.883

Abstract

This work concerns asymptotical stabilisation phenomena occurring in the moduli space of sections of certain algebraic families over a smooth projective curve, whenever the generic fibre of the family is a smooth projective Fano variety, or not far from being Fano.

We describe the expected behaviour of the class, in a ring of motivic integration, of the moduli space of sections of given numerical class. Up to an adequate normalisation, it should converge, when the class of the sections goes arbitrarily far from the boundary of the dual of the effective cone, to an effective element given by a motivic Euler product. Such a principle can be seen as an analogue for rational curves of the Batyrev–Manin–Peyre principle for rational points.

The central tool of this article is the property of equidistribution of curves. We show that this notion does not depend on the choice of a model of the generic fibre, and that equidistribution of curves holds for smooth projective split toric varieties. As an application, we study the Batyrev–Manin–Peyre principle for curves on a certain kind of twisted products.

Keywords

Manin's conjectures, rational curves, motivic Euler products, toric varieties, equidistribution, twisted products, motivic stabilisation

Mathematical Subject Classification

Primary: 14H10, 14E18, 11G50

Secondary: 11M41, 14G40

Milestones

Received: 31 March 2023

Revised: 30 May 2024

Accepted: 16 July 2024

Published: 22 April 2025

Authors

Loïs Faisant
Institute of Science and Technology Austria Klosterneuburg Austria

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Vol. 19, No. 5, 2025