The points of a cubic irreducible hypersurface of \({\mathbb{P}}^ n_ k\) as complete intersection subset
type: Item
instance of: scholarly article
title: The points of a cubic irreducible hypersurface of \({\mathbb{P}}^ n_ k\) as complete intersection subset
zbMATH Open document ID: 0537.14031
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1983-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/107920
review text: Let F be an irreducible reduced cubic hypersurface of \({\mathbb{P}}^ n_ k\), \(n\geq 3\), defined over an algebraically closed field k of char k\(=2\) and assume that F is not a cone whose vertex has dimension n-3. The main result states that through any point \(p\in F\) there passes a plane curve \(C\subset {\mathbb{P}}^ n\) with deg \(C\leq 4\) such that \(F\cdot C=(3\cdot \deg C)p.\) Since C is a complete intersection this means that every point of F is set-theoretically complete intersection (s.t.c.i.), being defined in the homogeneous coordinate ring of F by an ideal with n-1 generators. This fact together with a result of \textit{E. Stagnaro} [Atti Accad. Ligure Sci. Lett. 31 (1974), 138-148 (1975; Zbl 0363.14014)] implies the projective classification of cubic surfaces of \({\mathbb{P}}^ 3_ k\) on which any algebraic subvariety is s.t.c.i. The author also shows that the main result of the paper fails for a cubic cone whose vertex has dimension n-3 in case the ground field k is uncountable with char k\(=0\).
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 3853267
zbMATH Keywords: set-theoretically complete intersection
zbMATH Keywords: projective classification of cubic surfaces
author: Pier Carlo Craighero
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
cites work: Punti di tipo 9 di una cubica ellittica
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description: scientific article; zbMATH DE number 3853267