On the classification of non-normal complete intersections of two quadrics
type: Item
instance of: scholarly article
title: On the classification of non-normal complete intersections of two quadrics
zbMATH Open document ID: 1246.14070
author: Wanseok Lee
author: Euisung Park
author: Peter Schenzel
published in: Journal of Pure and Applied Algebra
publication date: 2012-07-30T00:00:00Z
Mathematics Subject Classification ID: 14N25
Mathematics Subject Classification ID: 51N35
zbMATH DE Number: 6060943
zbMATH Keywords: del Pezzo varieties
zbMATH Keywords: complete intersections
zbMATH Keywords: varieties of minimal degree
Wikidata QID: Q112881999
reviewed by: Roberto Muñoz
describes a project that uses: SINGULAR
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.12.009
OpenAlex ID: W2035113077
cites work: On varieties of almost minimal degree in small codimension
cites work: Arithmetic properties of projective varieties of almost minimal degree
cites work: Q3995221
cites work: Q4002797
cites work: On the classification of non-normal cubic hypersurfaces
cites work: An Enumeration of All Varieties of Degree 4
DOI: 10.1016/J.JPAA.2011.12.009
review text: 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.
description: scientific article; zbMATH DE number 6060943