m at h . A G ] 1 6 Ju n 20 02 ON A CLASSICAL CORRESPONDENCE BETWEEN K 3 SURFACES

Let X be a K3 surface which is intersection of three (a net P) of quadrics in P. The curve of degenerate quadrics has degree 6 and defines a double covering of P K3 surface Y ramified in this curve. This is the classical example of a correspondence between K3 surfaces which is related with moduli of vector bundles on K3 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 19-dimensional moduli of intersections of three quadrics in P when Y ∼= X. One of them when X has a line is classical and corresponds to d = 17. Similar considerations can be applied for a realization of an 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.