On the classification of non-normal complete intersections of two quadrics
A nondegenerate irreducible projective variety \(X \subset \mathbb{P}^r\) is said a \textit{del Pezzo} variety if the difference between its degree and its codimension is two, being arithmetically Cohen-Macaulay. It is known that a non-normal del Pezzo variety is the image by a linear projection (from a point) of a variety of minimal degree and that its singular locus is a linear space of codimension one in \(X\). These facts allowed the authors of this paper to classify all non-normal cubic hypersurfaces up to projective equivalence. In the paper under review they consider the next case: varieties of degree 4 and codimension two, which in fact have to be, being Cohen-Macaulay, the complete intersection of two quadrics. The classification result of non-normal complete intersections \(X \subset \mathbb{P}^r\) of two quadrics \(Q_1\) and \(Q_2\) is stated in two theorems (Theorem 1.1 and 1.2) where the complete list of non equivalent \(X \subset \mathbb{P}^r\) is presented: they are of 6 types if the characteristic of the base field is different from two and of three extra types in characteristic two.NEWLINENEWLINEThe proof consists of two parts: first, use the very special form of the singular locus of \(X\) to transform \(Q_1\) and \(Q_2\) in a much simpler form, bounding in particular \(r\) and leading to one of the types of the list; second, prove that the elements in the list are not projectively equivalent by a careful study of its normalization map, the projection of a minimal degree variety.
scientific article; zbMATH DE number 6060943