We present a method to compute the geometric Picard rank of a K3 surface over É. Contrary to a widely held belief, we show it is possible to verify Picard rank 1 using reduction only at a single prime. Our method is based on deformation theory for invertible sheaves.
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 P 3 . Using the theory of abelian varieties, the Picard group may be computed directly in this case. Our experiments show that the upper bounds provided by 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 rk Pic(V Fp ) 20 for a set of primes of density at least 1/2.
Abstract:We present a method to construct non-singular cubic surfaces over Q with a Galois invariant double-six.We start with cubic surfaces in the hexahedral form of L. Cremona and Th. Reye. For these, we develop an explicit version of Galois descent.
MSC:14J26, 14G25
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