Motivic distribution of rational curves and twisted products of toric varieties

TL;DR

This paper explores the asymptotic behavior of the moduli space of rational curves on algebraic families over curves, demonstrating equidistribution on toric varieties and proposing a motivic analogue of the Batyrev-Manin-Peyre principle.

Contribution

It introduces a motivic framework for understanding the distribution of rational curves, proving equidistribution for toric varieties and analyzing twisted products.

Findings

Moduli space classes converge to a motivic Euler product.

Equidistribution of curves holds for smooth projective split toric varieties.

The motivic principle parallels the Batyrev-Manin-Peyre conjecture for rational points.

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.