Kummer surfaces and the computation of the Picard group

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 ${\text{bP}}^3$. 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 $\text{rk}\text{Pic}(V_{{\overline{\text{bbF}}}_p})\ge 20$ for a set of primes of density $\ge 1\mathrm{/}2$.