Let \(P\) be a generic net of quadrics in \(\mathbb{P}^5\). Its spectral curve is a smooth sextic curve \(C\subset P\cong\mathbb{P}^2\), and the double covering \(\hat S\to P\) ramified at \(C\) is a \(K3\)-surface. On the other hand, the fundamental locus of \(P\) is also a \(K3\)-surface (triquadric) \(S\subset\mathbb{P}^5\); clearly, \(S\) determines \(P\). The assignment \(S\to\hat S\) is called in the paper the classical correspondence for \(K3\)-surfaces, see, e.g., [C. Madonna and V. V. Nikulin, Proc. Steklov Inst. Math. 241, 120–153 (2003; Zbl 1076.14046); Izv. Math. 72, No. 3, 497–508 (2008; Zbl 1162.14025)].
The author studies this correspondence in the case where \(P\) (and, hence, also \(S\) and \(\hat S\)) are real. One of the principal results is Theorem 0.1 stating that {\(S\) and \(\hat S\) are deformation equivalent in the class of abstract real \(K3\)-surfaces if and only if \([\mathbb{R} H]=0\) in \(H_1(\mathbb{R} S;\mathbb{F}_2)\)}, where \(H\) is a generic real hyperplane section of \(S\). More generally, the author determines the deformation class of \(\hat S\) (in the form of its Nikulin’s invariants, see [V. V. Nikulin, Math. USSR, Izv. 14, 103–167 (1980; Zbl 0427.10014)]) in terms of that of \(S\); a list of the invariants of \(S\) and corresponding invariants of \(\hat S\) is the contents of Theorem 0.2.