Between K 3 Surfaces

Let X be a K3 surface which is intersection of three (i.e. a net P) of quadrics in P. The curve of degenerate quadrics has degree 6 and defines a natural double covering of P ramified in this curve which is again a K3. This is a classical example of a correspondence between K3 surfaces which is related with moduli of sheaves on K3’s studied by Mukai. When general (for fixed Picard lattices) X and Y are isomorphic? We give necessary and sufficient conditions in terms of Picard lattices of X and Y . E.g. for Picard number 2 the Picard lattice of X and Y is defined by its determinant −d where d > 0, d ≡ 1 mod 8, and one of equations a − db = 8 or a − db = −8 should have an integral solution (a, b). Clearly, the set of these d is infinite: d ∈ {(a ∓8)/b} where a and b are odd integers. This describes all possible divisorial conditions on the 19-dimensional moduli of intersections of three quadrics in P with Y ∼= X. One of them, when X has a line is classical and corresponds to d = 17. Similar considerations can be applied to a realization of any isomorphism (T (X)⊗ Q, H(X)) ∼= (T (Y ) ⊗ Q, H(Y )) of transcendental periods over Q of two K3 surfaces X and Y by a fixed sequence of types of Mukai vectors.