On quintic \(K\)-3 surfaces (Q1175593)

Irreducible quintic surfaces with one-dimensional singular loci in the three-dimensional projective space are studied in this paper under the condition that they are birationally isomorphic to \(K3\) surfaces. In the first part three kinds of methods to construct such quintic surfaces starting from smooth \(K3\) surfaces are given. In the second part it is shown that conversely quintic surfaces satisfying the conditions can be constructed by one of methods in the first part under certain generic conditions. These generic conditions are not so clear. It is an interesting problem to eliminate these conditions. Besides, it may be interesting to consider further whether a quintic surface with a singular locus along a line and with a double point outside this line can be birationally isomorphic to a \(K3\) surface.