Three constructions of rationality of a four-dimensional cubic (Q1063084)

Let X be a four-dimensional cubic hypersurface in \({\mathbb{C}}P^ 5\). If X is general it is not known whether X is a rational variety. In the paper under review the author discusses three constructions for some special such cubic hypersurfaces X in order to prove their rationality. These special cubics are: (i) smooth cubics X containing two non-intersecting planes in \(P^ 5\), (ii) smooth cubics containing the image of \(P^ 1\times P^ 1\hookrightarrow P^ 5\) given by the linear system \(| {\mathcal O}(2,1)|\), and (iii) smooth cubics containing a Del Pezzo surface of degree 5 in \(P^ 5\).