2002/06/16 by Carlo Madonna, Madonna, Carlo, Viacheslav V. Nikulin +1 · 2 citations
Mathematics · Arts and Humanities · #Algebraic Geometry and Number Theory #Historical Studies and Socio-cultural Analysis #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.math/0206158
Let X be a K3 surface which is intersection of three (a net P2) of quadrics in P5. The curve of degenerate quadrics has degree 6 and defines a double covering of P2 K3 surface Y ramified in this curve. This is a 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 a2-db2=8 or a2-db2=-8 should have an integral solution (a,b). The set of these d is infinite: d∈ (a2∓ 8)/b2 where a and b are odd integers. This describes all possible divisorial conditions on 19- dimensional moduli of intersections of three quadrics in P5 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, H2,0(X)) ≅ (T(Y)⊗ Q, H2,0(Y)) of transcendental periods over Q of two K3 surfaces X and Y by a fixed sequence of types of Mukai vectors.