On the classification of non-normal cubic hypersurfaces
The main results of this article deal with non-normal, projective, cubic hypersurfaces over an algebraically closed field \(K\). If the ambient projective space has dimension at least \(5\), then such a hypersurface is necessarily a cone (which follows from results proved elsewhere). Thus the problem of determining the projective equivalence classes of such hypersurfaces reduces to the one in \(4\) or lower dimensional projective spaces over \(K\). In Theorem 3.1 of the article, authors provide a list of explicit canonical equations of (non-normal, projective, cubic) hypersurfaces corresponding to the distinct projective equivalence classes. As to be expected, when \(K\) has characteristic \(2\) or \(3\) there are more classes. The authors' proof proceeds via a detailed description of the normalization of the hypersurfaces defined by the canonical equations. The ensuing classification is asserted in Theorem 4.2. Normalization is used only to establish the projective inequivalence of the equations listed in Theorem 3.1.
scientific article; zbMATH DE number 5886360