[Paper Review] A practical algorithm to compute the geometric Picard lattice of K3 surfaces of degree $2$

Dino Festi|arXiv (Cornell University)|Aug 1, 2018

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper presents a practical algorithm to compute the geometric Picard lattice of K3 surfaces of degree 2 over number fields or function fields over Q, using a four-step process involving upper bounds on Picard rank, explicit divisor computation, lattice generation, and verification via mod-p Galois representations. The algorithm may not terminate but, if it does, returns the correct full Picard lattice; even if truncated, the output remains a provably correct sublattice useful for geometric and arithmetic applications.

ABSTRACT

Let $k$ be either a number a field or a function field over $\mathbb{Q}$ with finitely many variables. We present a practical algorithm to compute the geometric Picard lattice of a K3 surface over $k$ of degree $2$, i.e., a double cover of the projective plane over $k$ ramified above a smooth sextic curve. The algorithm might not terminate, but if it terminates then it returns a proven correct answer.

Motivation & Objective

• To develop a practical, implementable algorithm for computing the geometric Picard lattice of K3 surfaces of degree 2 over number fields or function fields over Q.
• To address the lack of existing algorithms that have been successfully used in practice or implemented in computer algebra systems.
• To provide a method that, even if terminated early, yields a provably correct sublattice of the geometric Picard lattice for applications such as detecting elliptic fibrations or rational points.
• To handle cases where the Picard lattice is not generated solely by −2-curves or hyperplane sections, by incorporating automorphism and Galois group actions.

Proposed method

• Step I: Compute an upper bound τ on the geometric Picard rank ρ(X̄) using van Luijk’s method or its refinements.
• Step II: Search for explicit algebraic curves (divisors) on the surface, particularly rational curves (−2-curves) and the hyperplane section.
• Step III: Generate the sublattice Λ ⊆ Pic(X̄) from the classes of the divisors found in Step II.
• Step IV: Verify whether Λ = Pic(X̄) using mod-p Galois representations and group actions via Proposition 7.7, checking whether elements in Λ_p lie in the kernel of ι_p.
• Use the action of the Galois group Gal(K/k) and automorphism group G to analyze orbits in Λ_p and rule out non-trivial kernel elements.
• Apply time constraints in practice: if the algorithm is stopped early, it still returns a sublattice Λ with a warning that it may not be full, but remains valid for partial applications.

Experimental results

Research questions

• RQ1 Can a practical algorithm be constructed to compute the geometric Picard lattice of a K3 surface of degree 2 over a number field or function field over Q?
• RQ2 Under what conditions does the algorithm fail to terminate, and what are the main obstructions to proving that the computed sublattice is the full Picard lattice?
• RQ3 Can the algorithm still yield useful results even when it is forced to terminate early, and what guarantees can be given about the correctness of the partial output?
• RQ4 To what extent can the Picard lattice be generated by −2-curves and the hyperplane section, and what are the implications when this is not the case?
• RQ5 How can mod-p Galois representations and group actions be used to verify the fullness of a candidate Picard lattice?

Key findings

• The algorithm is designed to be practical and has already been successfully used in practice, as noted in the cited reference [1].
• Even if the algorithm does not terminate, the output sublattice Λ is provably contained in the geometric Picard lattice Pic(X̄), making it useful for applications such as detecting elliptic fibrations or potential density of rational points.
• The algorithm may fail to terminate due to a non-sharp upper bound τ > ρ(X̄), which causes endless attempts to find more divisors.
• The algorithm may also fail if the geometric Picard lattice cannot be generated by −2-curves and the hyperplane section, as the search for such curves may not terminate.
• Even when Λ = Pic(X̄), the verification step (Step IV) may fail to detect this due to insufficiently large orbits under G or Gal(K/k), leading to infinite loops.
• In cases where the algorithm is truncated, the output remains a valid sublattice, and the user is warned that it may not be full, but it can still be used for partial arithmetic and geometric analysis.