Kummer surfaces and the computation of the Picard group

TL;DR

This paper evaluates R. van Luijk's method for computing the Picard group of K3 surfaces, specifically Kummer quartics, demonstrating its effectiveness and limitations through experimental analysis.

Contribution

The paper provides an empirical assessment of van Luijk's method on Kummer quartics, highlighting its accuracy and the behavior of Picard ranks over various primes.

Findings

Van Luijk's bounds are sharp with enough primes.

Many Kummer surfaces have Picard rank ≥20 at certain primes.

The method's accuracy varies depending on the surface and primes used.

Abstract

We test R. van Luijk's method for computing the Picard group of a $K3$ surface. The examples considered are the resolutions of Kummer quartics in $\bP^3$. Using the theory of abelian varieties, in this case, the Picard group may be computed directly. Our experiments show that the upper bounds provided by R. van Luijk's method are sharp when sufficiently many primes are used. In fact, there are a lot of primes that yield a value close to the exact one. However, for many but not all Kummer surfaces $V$ of Picard rank 18, we have $\smash{\rk\Pic(V_{\bar\bbF_{p}}) \geq 20}$ for a set of primes of density $\geq 1/2$.