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Motivic distribution of rational curves and twisted products of toric varieties

Loïs Faisant

TL;DR

The paper develops a geometric, motivic analogue of the Batyrev-Manin-Peyre principle for rational curves on Fano-like varieties, framed in the Grothendieck ring of varieties and augmented by motivic Euler products and weight filtrations. It introduces and analyzes moduli spaces of sections over a fixed curve, equipped with multidegrees relative to chosen Picard-basis models, and proves a key equidistribution principle that is invariant under changes of model. This leads to a motivic Tamagawa number, expressed as a motivic Euler product, and convergence results for toric and twisted-product contexts, including explicit formulas in the toric case with Möbius-type decompositions. The framework unifies arc-space techniques (Greenberg schemes), weak Néron models, and motivic integration to connect asymptotics of moduli spaces with Tamagawa-like constants, enabling the study of twisted products of toric varieties and their rational curves in characteristic-free settings.

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.

Motivic distribution of rational curves and twisted products of toric varieties

TL;DR

The paper develops a geometric, motivic analogue of the Batyrev-Manin-Peyre principle for rational curves on Fano-like varieties, framed in the Grothendieck ring of varieties and augmented by motivic Euler products and weight filtrations. It introduces and analyzes moduli spaces of sections over a fixed curve, equipped with multidegrees relative to chosen Picard-basis models, and proves a key equidistribution principle that is invariant under changes of model. This leads to a motivic Tamagawa number, expressed as a motivic Euler product, and convergence results for toric and twisted-product contexts, including explicit formulas in the toric case with Möbius-type decompositions. The framework unifies arc-space techniques (Greenberg schemes), weak Néron models, and motivic integration to connect asymptotics of moduli spaces with Tamagawa-like constants, enabling the study of twisted products of toric varieties and their rational curves in characteristic-free settings.

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.
Paper Structure (54 sections, 46 theorems, 490 equations, 1 figure)

This paper contains 54 sections, 46 theorems, 490 equations, 1 figure.

Table of Contents

  1. Models and moduli spaces of sections
  2. Global models and degrees
  3. Moduli spaces of curves
  4. Greenberg schemes and motivic integrals
  5. Local intersection degrees
  6. Weak Néron models and smoothening
  7. Local models
  8. From local models to global ones
  9. Piecewise trivial fibrations
  10. Motivic Euler products
  11. The weight filtration
  12. Mixed Hodge modules
  13. The Hodge realisation of $K_0 \mathbf{Var}_{S}$
  14. The weight filtration on $K_0 \mathbf{Var}_{S}$
  15. Useful examples and vanishing properties
  16. ...and 39 more sections

Key Result

Theorem A

If $\mathscr{B}$ is a Fano-like variety over $k$ verifying the motivic Batyrev-Manin-Peyre principle for curves, $X$ is a smooth projective split toric variety over $k$, with torus $T$, and $\mathscr{T}$ is a Zariski-locally trivial $T$-torsor over $\mathscr{B}$, then the twisted product verifies the motivic Batyrev-Manin-Peyre principle as well: the answer to BMP-motivic-weak is positive for rat

Figures (1)

  • Figure :

Theorems & Definitions (136)

  • Definition 1
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem A: \ref{['thm:BMP-twisted-products']}
  • Theorem B: \ref{['thm-equidistribution-toric']}
  • Theorem C: \ref{['thm:equidistribution-and-models']}
  • Example 1.1
  • Remark 1.2
  • ...and 126 more