Simple K3 singularities which are hypersurface sections of toric singularities (Q1187166)

The author investigates hypersurface sections in four-dimensional affine toric varieties. Let a non-degenerate function \(f\in\mathbb{C}[\check\sigma\cap\mathbb{Z}^ 4]\) be given \((\sigma\subset\mathbb{R}^ 4\) is a polyhedral cone, \(\check\sigma\) denotes its dual). If \(f\) defines a simple K3 singularity \([f=0]\subseteq\text{Spec} \mathbb{C}[\check\sigma\cap\mathbb{Z}^ 4]\) then \(\sigma\) has to be Gorenstein and the Newton polyhedron \(\Gamma_ +(f)\subset\check\sigma\) of \(f\) will meet a very special condition: It has to be contained in some suitable affine hyperplane (defined by an element \(u_ 0\in int(\sigma))\) meeting special conditions. --- Finally, it is proved that for such pairs \((\sigma,u_ 0)\) there are (up to automorphisms) only finitely many possibilities to occur. An algorithm to determine these pairs is given also.