Determinantal quartics and the computation of the Picard group

We test the methods for computing the Picard group of a $K3$ surface in a situation of high rank. The examples chosen are resolutions of quartics in $\bP^3$ having 14 singularities of type $A_1$. Our computations show that the method of R. van Luijk works well when sufficiently large primes are used.

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