On a map between two \(K3\) surfaces associated to a net of quadrics (Q868649)

The authors study a net \(N\) of quadrics containing a line. Let \(S\) be the intersection of the net and \(S_6\) the double cover of \(N \cong \mathbb{P}^2\) branched along the discriminant of the net (which is a sextic). Then it is known that both \(S\) and \(S_6\) are birational to \(K3\) surfaces and there is a birational map between them. The authors show (among other things): (i) \(S_6\) has only type \(A_1\) singularities (corresponding to the rank-4 quadrics in the net); (ii) \(S\) is the blowup of \(S_6\) at singularities, and (iii) the exceptional locus of \(S \to S_6\) consists of lines, conics and twisted cubics. The authors further give an interpretation of the map \(S \to S_6\) in terms of classical algebraic geometry and investigate the relation between the lines the branch divisors. Various examples are constructed. Results of \textit{C. Madonna} and \textit{V. V. Nikulin} [Proc. Steklov Math. Inst. 241, 120--153 (2003; Zbl 1076.14046) and ``On a classical correspondence between \(K3\) surfaces''. II, preprint, \url{arXiv:math.AG/0304415}] are applied.