Let P denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z, and w; let X be the generic element of the family of surfaces in P given byThe surface X is a K3 surface over the function field Q(t). In this paper, we explicitly compute the geometric Picard lattice of X , together with its Galois module structure, as well as derive more results on the arithmetic of X and other elements of the family X.
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