Real K3 Surfaces with non-symplectic Involutions and Applications

Classification of real K3 surfaces X with a non-symplectic involution \tau is considered. For some exactly defined and one of the weakest possible type of degeneration (giving the very reach discriminant), we show that the connected component of their moduli is defined by the isomorphism class of the action of \tau and the anti-holomorphic involution \phi in the homology lattice. (There are very few similar cases known.) For their classification we apply invariants of integral lattice involutions with conditions which were developed by the first author in 1983. As a particular case, we describe connected components of moduli of real non-singular curves A\in |-2K_V| for the classical real surfaces: V=P^2, hyperboloid, ellipsoid, F_1, F_4. As an application, we describe all real polarized K3 surfaces which are deformations of general real K3 double rational scrolls (surfaces V above). There are very few exceptions. For example, any non-singular real quartic in P^3 can be constructed in this way.