On set-theoretic intersections and connectedness of algebraic varieties in \(\mathbb P^n\)
type: Item
instance of: scholarly article
title: On set-theoretic intersections and connectedness of algebraic varieties in \(\mathbb P^n\)
zbMATH Open document ID: 0649.14026
DOI: 10.1007/BF01194099
author: Uwe Nagel
author: Wolfgang Vogel
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: The aim of this note is to show that the use of connectedness does not help so much in estimating the minimum number \(\text{ara}(V)\) of equations which are needed to define a projective variety \(V\) set-theoretically, even in the case that \(V\) is defined by a monomial ideal. It is known that if \(V\subset\mathbb P^n_k\) is connected in codimension \(k\), but not in codimension \(k-1\), and if \(\text{ara}(V)<n\), then \(ara(V)\leq \text{codim} (V)+k-1\) [see e.g. \textit{J. Rung}, ``Mengentheoretische Durchschnitte und Zusammenhang'', Regensburger Math. Schr. 3 (1979; Zbl 0512.14030)]. This note gives a class of examples where \(V\) is defined by a monomial ideal and the difference \(\text{ara}(V)-(\text{codim} (V)+k-1)\) can be arbitrarily large.
reviewed by: Trung Ngo Viet
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 54D05
zbMATH DE Number: 4059581
zbMATH Keywords: set-theoretical complete intersection
zbMATH Keywords: connectedness
zbMATH Keywords: monomial ideal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4059581