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Explicit desingularisation of Kummer surfaces in characteristic two via specialisation

Alvaro Gonzalez-Hernandez

TL;DR

The paper develops explicit, field-defined models for the Jacobians of genus two curves and their Kummer surfaces in characteristic two by specializing characteristic-zero constructions. It provides an end-to-end computational framework to obtain Jacobian and Kummer embeddings (Jacobian in $\mathbb{P}^{15}$, Kummer in $\mathbb{P}^{3}$ and a partial desingularisation in $\mathbb{P}^{5}$), together with a detailed analysis of tropes, $2$-torsion actions, and the Weddle surface geometry. The work distinguishes ordinary, almost ordinary, and supersingular cases in characteristic two, describing the singularities and resolutions, and gives an explicit example of a Kummer surface with everywhere good reduction over a quadratic field, along with a Lazda–Skorobogatov criterion analysis. It also provides extensive appendices with characteristic-two formulae and connections to Katsura–Kondō models, enabling practical computation of desingularisations directly in characteristic two. Overall, the results demonstrate that characteristic-two behavior of Kummer surfaces can be studied via specialization from characteristic zero, yielding concrete models and reduction criteria with broad arithmetic implications.

Abstract

We study the birational geometry of the Kummer surfaces associated to the Jacobian varieties of genus two curves, with a particular focus on fields of characteristic two. In order to do so, we explicitly compute a projective embedding of the Jacobian of a general genus two curve and, from this, we construct its associated Kummer surface. This explicit construction produces a model for desingularised Kummer surfaces over any field of characteristic not two, and specialising these equations to characteristic two provides a model of a partial desingularisation. Adapting the classic description of the Picard lattice in terms of tropes, we also describe how to explicitly find completely desingularised models of Kummer surfaces whenever the $p$-rank is not zero. In the final section of this paper, we compute an example of a Kummer surface with everywhere good reduction over a quadratic number field, and draw connections between the models we computed and a criterion that determines when a Kummer surface has good reduction at two.

Explicit desingularisation of Kummer surfaces in characteristic two via specialisation

TL;DR

The paper develops explicit, field-defined models for the Jacobians of genus two curves and their Kummer surfaces in characteristic two by specializing characteristic-zero constructions. It provides an end-to-end computational framework to obtain Jacobian and Kummer embeddings (Jacobian in , Kummer in and a partial desingularisation in ), together with a detailed analysis of tropes, -torsion actions, and the Weddle surface geometry. The work distinguishes ordinary, almost ordinary, and supersingular cases in characteristic two, describing the singularities and resolutions, and gives an explicit example of a Kummer surface with everywhere good reduction over a quadratic field, along with a Lazda–Skorobogatov criterion analysis. It also provides extensive appendices with characteristic-two formulae and connections to Katsura–Kondō models, enabling practical computation of desingularisations directly in characteristic two. Overall, the results demonstrate that characteristic-two behavior of Kummer surfaces can be studied via specialization from characteristic zero, yielding concrete models and reduction criteria with broad arithmetic implications.

Abstract

We study the birational geometry of the Kummer surfaces associated to the Jacobian varieties of genus two curves, with a particular focus on fields of characteristic two. In order to do so, we explicitly compute a projective embedding of the Jacobian of a general genus two curve and, from this, we construct its associated Kummer surface. This explicit construction produces a model for desingularised Kummer surfaces over any field of characteristic not two, and specialising these equations to characteristic two provides a model of a partial desingularisation. Adapting the classic description of the Picard lattice in terms of tropes, we also describe how to explicitly find completely desingularised models of Kummer surfaces whenever the -rank is not zero. In the final section of this paper, we compute an example of a Kummer surface with everywhere good reduction over a quadratic number field, and draw connections between the models we computed and a criterion that determines when a Kummer surface has good reduction at two.
Paper Structure (31 sections, 9 theorems, 90 equations, 6 tables)

This paper contains 31 sections, 9 theorems, 90 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Projective models of Kummer surfaces in characteristic not two
  3. Translation by a 2-torsion point
  4. Tropes of a Kummer surface
  5. Kummer surfaces over fields of characteristic two
  6. Computing models of Jacobian and Kummer surfaces
  7. Computing a basis of $\mathcal{L}(2(\Theta_++\Theta_-))$
  8. Computing the equations of the Jacobian
  9. Computing models of Kummer surfaces and its desingularisation in characteristic two
  10. Partial desingularisations of Kummer surfaces in characteristic two
  11. The geometry of the ordinary case
  12. The geometry of the almost ordinary case
  13. The geometry of the supersingular case
  14. Weddle surfaces and the blow-ups of the exceptional lines
  15. The ordinary case
  16. ...and 16 more sections

Key Result

Theorem 1.1

Given a curve of genus two over a perfect field of characteristic two, we can compute an explicit projective embedding of its Jacobian as the intersection of $72$ quadrics in $\mathbb{P}^{15}$. We can also compute a projective embedding for a partial desingularisation of its associated Kummer surfac

Theorems & Definitions (12)

  • Theorem 1.1
  • Proposition 2.1: Flynn2012Two-coverings2
  • Proposition 2.2: Flynn2012Two-coverings2
  • Proposition 2.3: Flynn2012Two-coverings2
  • Theorem 3.1
  • Proposition 5.1
  • proof
  • Proposition 6.1
  • proof
  • Theorem 7.1
  • ...and 2 more