Semistable \(K3\)-surfaces with icosahedral symmetry. (Q1419570)

A degeneration of \(K3\)-surfaces is a \(1\)-parameter family with general fibre a smooth \(K3\)-surface. After a ramification of the base and resolution of singularities one may assume that the degeneration \(f: {\mathcal X} \to S \ni 0\) is semistable, that is, the central fibre \(X=f^{-1}(0)\) is a reduced divisor with simple normal crossing in the smooth manifold \({\mathcal X}\). It is a well known result of \textit{V. S. Kulikov} [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 1115--1119 (1980; Zbl 0463.14011)] that semistable degenerations of \(K3\)-surfaces, \(f: {\mathcal X} \to S\), such that all components of the central fibre \(X=f^{-1}(0)\) are Kähler admit a modification \({\mathcal X}'\) of \({\mathcal X}\) such that \(K_{\mathcal{X}'}\equiv 0.\) Such a degeneration is called a Kulikov model. It is also proved, see Kulikov [loc. cit.], \textit{U. Persson} [``On degenerations of algebraic surfaces'', Mem. Am. Math. Soc. 11, No. 189 (1977; Zbl 0368.14008)], that if \(f: {\mathcal X} \to S\) is a Kulikov model of a degeneration of \(K3\)-surfaces such that all components of \(X=f^{-1}(0)\) are Kähler then the central fibre \(X\) can be of three types. According to these three types one speaks of degenerations of type I, II, or III. In the paper under review the author considers type III degenerations of \(K3\)-surfaces and precisely those in which the dual graph of the central fibre is a triangulation of \(S^2\). In particular the author realizes tetrahedral, octahedral and icosahedral triangulation in families of \(K3\)-surfaces.