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Motivic counting of rational curves with tangency conditions via universal torsors

Loïs Faisant

TL;DR

The paper studies the Grothendieck motive of the moduli space $\mathrm{Hom}_k(\mathscr{C},X)$ for a smooth curve $\mathscr{C}$ and a Mori Dream Space $X$, using Cox rings and universal torsors to obtain an explicit parametrisation. It proves an explicit algebraic description of morphisms with images meeting a fixed open set $U$ in $X$ and, when $X$ is a smooth split toric variety, establishes a motivic Batyrev–Manin–Peyre principle for curves satisfying tangency conditions (Campana curves) with respect to the boundary divisors. The key technical framework rests on representing $\mathrm{Cox}(X)$, constructing universal torsors, and translating morphisms into $X$-collections subject to coprimality and linear relations; these lead to a decomposition of the motivic counts as motivic Euler products. The results generalise previous motivic BMP instances from $\mathscr{C}=\mathbf{P}^1_k$ to arbitrary curves and provide a motivic distribution picture for Campana curves on toric varieties, with potential applications to function-field analogues of Manin’s conjecture and to motivic Tamagawa numbers. Overall, the work advances a unifying, Cox-ring–torsor approach to counting morphisms and their motivic invariants in an isotrivial setting, including tangency constraints.

Abstract

Using the formalism of Cox rings and universal torsors, we prove a decomposition of the Grothendieck motive of the moduli space of morphisms from an arbitrary smooth projective curve to a Mori Dream Space (MDS). For the simplest cases of MDS, that of toric varieties, we use this decomposition to prove an instance of the motivic Batyrev--Manin--Peyre principle for curves satisfying tangency conditions with respect to the boundary divisors, often called Campana curves.

Motivic counting of rational curves with tangency conditions via universal torsors

TL;DR

The paper studies the Grothendieck motive of the moduli space for a smooth curve and a Mori Dream Space , using Cox rings and universal torsors to obtain an explicit parametrisation. It proves an explicit algebraic description of morphisms with images meeting a fixed open set in and, when is a smooth split toric variety, establishes a motivic Batyrev–Manin–Peyre principle for curves satisfying tangency conditions (Campana curves) with respect to the boundary divisors. The key technical framework rests on representing , constructing universal torsors, and translating morphisms into -collections subject to coprimality and linear relations; these lead to a decomposition of the motivic counts as motivic Euler products. The results generalise previous motivic BMP instances from to arbitrary curves and provide a motivic distribution picture for Campana curves on toric varieties, with potential applications to function-field analogues of Manin’s conjecture and to motivic Tamagawa numbers. Overall, the work advances a unifying, Cox-ring–torsor approach to counting morphisms and their motivic invariants in an isotrivial setting, including tangency constraints.

Abstract

Using the formalism of Cox rings and universal torsors, we prove a decomposition of the Grothendieck motive of the moduli space of morphisms from an arbitrary smooth projective curve to a Mori Dream Space (MDS). For the simplest cases of MDS, that of toric varieties, we use this decomposition to prove an instance of the motivic Batyrev--Manin--Peyre principle for curves satisfying tangency conditions with respect to the boundary divisors, often called Campana curves.

Paper Structure

This paper contains 35 sections, 21 theorems, 219 equations, 1 figure.

Table of Contents

  1. Rings of varieties and motivic Euler products
  2. Monoids and rings of varieties
  3. Motivic euler products
  4. Convergence of motivic series
  5. Geometric realisation of coherent sheaves
  6. Flattening stratification
  7. Geometric realisation of coherent sheaves
  8. Picard schemes, schemes of linear systems and effective divisors
  9. Some functors and representability criterions
  10. Poincaré sheaves for the Picard scheme
  11. Motivic Euler products on $\mathscr{C}$ as motivic functions on $\mathop{\mathrm{\mathbf{Div}}}\nolimits_{\mathscr{C} / k}$
  12. Motivic Batyrev-Manin-Peyre principle
  13. Motivic Tamagawa numbers
  14. Motivic stabilisation of curves
  15. Cox rings and universal torsors of Mori Dream Spaces
  16. ...and 20 more sections

Key Result

Theorem A

As a scheme over the scheme $(\mathop{\mathrm{\mathbf{Div}}}\nolimits_{\mathscr{C} /k})^\mathfrak I$ parametrising $\mathfrak I$-tuples of effective divisors on $\mathscr{C}$, the parameter space is isomorphic to an explicit locally closed subset in given by the ideal $\mathcal{I}_X$ together with the coprimality conditions coming from the mutual non-intersections of the $\mathcal{D}_i$ for $i\i

Figures (1)

  • Figure 1: An example of a Campana curve on a split projective toric variety. Here, the image of $\mathscr{C}$ by $f$ is tangent to four of the prime boundary divisors.

Theorems & Definitions (63)

  • Theorem A: \ref{['thm-parametrisation-finale']} thm-parametrisation-finale
  • Theorem B: \ref{['thm-Campana-final']} thm-Campana-final
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • ...and 53 more