A practical algorithm to compute the geometric Picard lattice of K3 surfaces of degree $2$

Abstract

Let $k$ be either a number a field or a function field over $\mathbb{Q}$ with finitely many variables. We present a practical algorithm to compute the geometric Picard lattice of a K3 surface over $k$ of degree $2$, i.e., a double cover of the projective plane over $k$ ramified above a smooth sextic curve. The algorithm might not terminate, but if it terminates then it returns a proven correct answer.

A remark about an application of the algorithm to quartc surfaces with a node. Proposition 5.1 corrected. Acknowledgements and bibliography updated. Few typos corrected. 14 pages, 1 figure. Comments still welcome!