Complete intersection subvarieties of general hypersurfaces
type: Item
instance of: scholarly article
title: Complete intersection subvarieties of general hypersurfaces
zbMATH Open document ID: 0888.14018
author: Endre Szabó
published in: Pacific Journal of Mathematics
publication date: 1997-07-28T00:00:00Z
review text: A complete intersection curve \(C\) contained in a general hypersurface \(X\) of degree \(d>n\geq 1\) in \(\mathbb{P}^{n+1}\) is called nontrivial, if the equation of \(X\) is not part of a regular sequence defining \(C\), i.e. if \(C\) is not cut out by \(n-1\) hypersurfaces on \(X\). The main result of this paper is a classification of the possible types (i.e. the degrees of the equations defining \(C\)) of nontrivial complete intersection curves contained in a general hypersurface \(X\) of degree \(d\geq 6\). Furthermore, the author shows that (1) there are no nontrivial complete intersection surfaces on \(X\), (2) for \(d>2n-1\), there are no nontrivial complete intersection curves on \(X\), (3) for \(d>\frac32n+ \frac12\), all nontrivial complete intersection curves on \(X\) are lines, and (4) for \(d>n+2\), all nontrivial complete intersection curves on \(X\) are lines or conics. The proof is divided into two parts. In the first part, results about the regularity and the components of Hilbert schemes of projective spaces are used to reduce the questions to a certain inequality for Hilbert functions of complete intersections. Then the author gives a rather long and tedious inductive proof of that inequality.
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 1025786
zbMATH Keywords: Noether-Lefschetz theorem
zbMATH Keywords: nontrivial complete intersection curves contained in a general hypersurface
zbMATH Keywords: Hilbert schemes
zbMATH Keywords: Hilbert functions
reviewed by: Martin Kreuzer
MaRDI profile type: Publication
full work available at URL: pjm.1996.175.271
OpenAlex ID: W1865180625
DOI: 10.2140/PJM.1996.175.271
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description: scientific article; zbMATH DE number 1025786