Explicit desingularisation of Kummer surfaces in characteristic two via specialisation

Technical Summary: Explicit Desingularisation of Kummer Surfaces in Characteristic Two via Specialisation

Problem Statement
Kummer surfaces, defined as the quotient of an abelian surface by the involution $P \mapsto -P$, are classical examples of K3 surfaces. While their geometry over fields of characteristic not equal to two is well-understood—specifically, the existence of explicit models for their desingularisation as the intersection of three quadrics in $\mathbb{P}^5$—the situation in characteristic two is significantly more complex. In characteristic two, the 2-torsion subgroup of the abelian surface is smaller (isomorphic to $(\mathbb{Z}/2\mathbb{Z})^r$ where $r \le 2$), leading to fewer singular points but of higher complexity (e.g., $D_4$, $D_8$, or elliptic singularities). Previous attempts to construct explicit models for these surfaces in characteristic two, such as those by Müller, failed to generate smooth desingularised models via the standard intersection of quadrics. Furthermore, the behavior of Kummer surfaces in characteristic two differs fundamentally from other characteristics, raising questions about whether classical constructions can be adapted and whether examples of Kummer surfaces with everywhere good reduction exist over number fields.

Methodology
The author employs a strategy of specialisation from characteristic zero to characteristic two. The core methodology involves:

1. Explicit Embeddings: Computing a projective embedding of the Jacobian $J$ of a genus two curve $C$ as the intersection of 72 quadrics in $\mathbb{P}^{15}$. This is achieved by constructing a basis for the linear system $|2(\Theta_+ + \Theta_-)|$, where $\Theta$ represents theta divisors.
2. Basis Construction via Smith Normal Form: To handle the reduction modulo two, the author adapts a basis of odd functions from characteristic zero. Since direct reduction often leads to linear dependence due to the collapse of eigenvalues $\pm 1$ to $1$, an algorithm (Algorithm 1) is used to compute the Smith normal form of the relation matrix. This yields a modified basis that reduces well to characteristic two, ensuring the resulting elements remain linearly independent.
3. Partial Desingularisation: Using the reduced basis, the author constructs a rational map from the singular Kummer surface $X \subset \mathbb{P}^3$ to a surface $Y \subset \mathbb{P}^5$. This surface $Y$ is defined as the complete intersection of three quadrics. The map is an isomorphism outside the singular locus of $X$ and acts as a partial desingularisation.
4. Geometric Analysis by p-rank: The geometry of $Y$ and its singularities are analyzed separately for the three cases of the Jacobian's p-rank: ordinary ($r=2$), almost ordinary ($r=1$), and supersingular ($r=0$). The author utilizes the theory of "tropes" (special conics on the Kummer surface) and their behaviour under specialisation to describe the exceptional divisors and the resulting singularities of $Y$.
5. Good Reduction Criterion: To address the existence of Kummer surfaces with everywhere good reduction, the paper applies a criterion by Lazda and Skorobogatov. This involves analyzing the action of the absolute Galois group on the 2-torsion points of the Jacobian over a quadratic number field, specifically checking if the exact sequence of 2-torsion modules splits upon reduction.

Key Contributions and Results

• Explicit Models in Characteristic Two: The paper provides the first explicit equations for the Jacobian of a general genus two curve in characteristic two as the intersection of 72 quadrics in $\mathbb{P}^{15}$ (Theorem 4.1).
• Partial Desingularisation: It constructs a model for a partial desingularisation of the Kummer surface as the intersection of three quadrics in $\mathbb{P}^5$ (Theorem 3.1). The paper classifies the singularities of this model based on the p-rank:
  • Ordinary: $Y$ has 12 singularities of type $A_1$.
  • Almost Ordinary: $Y$ has 2 singularities of type $D_4^0$ and 2 of type $A_3$.
  • Supersingular: $Y$ has an elliptic singularity of type $A^*_{0} + A^*_{0} + A^*_{0} + A^*_{0} + A^*_{0}$ (in Laufer's notation).
• Geometric Description of Tropes: The paper extends the classical description of tropes to characteristic two, detailing how they specialise and how their intersections with exceptional divisors determine the singularity structure of the desingularised model.
• Existence of Everywhere Good Reduction: The paper proves the existence of Kummer surfaces with everywhere good reduction over a number field (Theorem 1.2). Specifically, it constructs an explicit example over the quadratic field $\mathbb{Q}(\sqrt{353})$ where the associated Kummer surface has good reduction at all primes, including those above 2. This is achieved by verifying that the Galois action on the 2-torsion points is compatible with the reduction, allowing the splitting of the relevant exact sequence without requiring a field extension for the specific example provided.
• Limitations and Extensions: The paper notes that while good reduction at 2 is possible, it is not guaranteed for all curves with everywhere good reduction. For "almost ordinary" cases, the minimal extension required to achieve good reduction may be of higher degree (e.g., degree 12), and for supersingular cases, the existence of a good reduction model is guaranteed by Matsumoto's theorem but the explicit extension is not provided.

Significance
The paper demonstrates that the classical geometric constructions of Kummer surfaces and their desingularisations, traditionally restricted to characteristic zero or odd characteristic, can be successfully adapted to characteristic two via specialisation. This bridges a gap in the understanding of K3 surfaces in positive characteristic, which is particularly relevant for cryptographic applications involving isogeny-based protocols (like SIDH) that often struggle in characteristic two. By providing explicit equations and a complete geometric classification of the singularities, the work enables computational verification and further study of these surfaces. Furthermore, the construction of a Kummer surface with everywhere good reduction over a quadratic field resolves a specific open question regarding the existence of such K3 surfaces, contrasting with the non-existence of abelian varieties with everywhere good reduction over $\mathbb{Q}$. The paper emphasizes that these results are constructive, supported by code that allows for the computation of explicit equations for the varieties described.