Real K3 surfaces with non-symplectic involution and applications. II

摘要

We consider real forms of rational surfaces F-m with Picard number 2. Connected components of moduli of real non-singular curves in vertical bar - 2K(Fm) vertical bar have been classified recently by us for m = 0, 1, 4. Applying similar methods, here we fill the gap for m = 2 and m = 3 to complete a similar classification for any 0 <= m <=, 4, when vertical bar -2K(Fm) vertical bar is reduced. The case of F-2 is especially remarkable and classical (the quadratic cone in P-3). As an application, we complete the classification of connected components of moduli of real hyper-elliptically polarized K3 surfaces and the classification of deformations of real hyper-elliptically polarized K3 surfaces to real polarized K3 surfaces started by us in 2005. This could be important in some questions because real hyper-elliptically polarized K3 surfaces can be constructed explicitly.