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

Abstract

We consider real forms of rational surfaces Fm with Picard number 2. Connected components of moduli of real non-singular curves in | − 2Fm| have been classified recently by us for m = 0, 1, 4. Applying similar methods, here we fill the gap for m = 2 andm = 3 to complete a similar classification for any 0 ⩽ m ⩽ 4, when | − 2Fm| is reduced. The case of F2 is especially remarkable and classical (the quadratic cone in ℙ3). 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.