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Iitaka fibrations and integral points: a family of arbitrarily polarized spherical threefolds

Ulrich Derenthal, Florian Wilsch

TL;DR

This work studies integral points on a family of spherical log Fano threefolds to realize Manin-type asymptotics in the integral setting and to illuminate logarithmic Iitaka fibrations. It combines Cox rings, universal torsors, and height machinery to reduce counting to fiberwise problems under the Iitaka fibration, distinguishing adjoint-rigid from moving cases. The authors compute archimedean and non-archimedean Tamagawa measures, alpha-constants, and fiberwise contributions, obtaining explicit leading constants and exponents and proving the conjectured form for this family. In rigid cases the fibration is constant; in moving cases the fibration is nontrivial and the coefficient is a sum of fiber-wise contributions, thus giving a complete, explicit, and sharp asymptotic formula with power-saving error terms in key regimes. Overall, the paper provides a logarithmic analogue of Iitaka fibrations for integral points and demonstrates the validity of Santens’ conjecture in a rich geometric setting of spherical log Fano threefolds.

Abstract

Studying Manin's program for a family of spherical log Fano threefolds, we determine the asymptotic number of integral points whose height associated with an arbitrary ample line bundle is bounded. This confirms a recent conjecture by Santens and sheds new light on the logarithmic analogue of Iitaka fibrations, which have not yet been adequately formulated.

Iitaka fibrations and integral points: a family of arbitrarily polarized spherical threefolds

TL;DR

This work studies integral points on a family of spherical log Fano threefolds to realize Manin-type asymptotics in the integral setting and to illuminate logarithmic Iitaka fibrations. It combines Cox rings, universal torsors, and height machinery to reduce counting to fiberwise problems under the Iitaka fibration, distinguishing adjoint-rigid from moving cases. The authors compute archimedean and non-archimedean Tamagawa measures, alpha-constants, and fiberwise contributions, obtaining explicit leading constants and exponents and proving the conjectured form for this family. In rigid cases the fibration is constant; in moving cases the fibration is nontrivial and the coefficient is a sum of fiber-wise contributions, thus giving a complete, explicit, and sharp asymptotic formula with power-saving error terms in key regimes. Overall, the paper provides a logarithmic analogue of Iitaka fibrations for integral points and demonstrates the validity of Santens’ conjecture in a rich geometric setting of spherical log Fano threefolds.

Abstract

Studying Manin's program for a family of spherical log Fano threefolds, we determine the asymptotic number of integral points whose height associated with an arbitrary ample line bundle is bounded. This confirms a recent conjecture by Santens and sheds new light on the logarithmic analogue of Iitaka fibrations, which have not yet been adequately formulated.
Paper Structure (20 sections, 10 theorems, 150 equations)

This paper contains 20 sections, 10 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Iitaka fibrations in the integral Manin problem
  3. Integral points on the spherical family
  4. Acknowledgements
  5. A family of spherical threefolds
  6. Cox rings and universal torsors
  7. Admissible boundaries
  8. Polarizations
  9. Height functions
  10. The leading constants
  11. Archimedean Tamagawa measures
  12. Non-archimedean Tamagawa measures
  13. Adelic Tamagawa measures
  14. Effective cones
  15. Moving adjoint divisors
  16. ...and 5 more sections

Key Result

Theorem 1.3

Let $\mathcal{X}$ be one of the polarized spherical log Fano threefolds described above. The number of integral points of bounded height satisfies for $B\ge 2$ with

Theorems & Definitions (24)

  • Conjecture 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • proof
  • ...and 14 more