Hypersurfaces of inifnite dimensional Banach spaces, Bertini theorems and embeddings of projective spaces
type: Item
instance of: scholarly article
title: Hypersurfaces of inifnite dimensional Banach spaces, Bertini theorems and embeddings of projective spaces
zbMATH Open document ID: 1050.32011
published in: Portugaliae Mathematica. Nova Série
publication date: 2004-05-18T00:00:00Z
full work available at URL: https://eudml.org/doc/51035
review text: This short paper in infinite dimensional algebraic geometry studies Bertini type problems and embeddings of projective spaces via clever methods based on hypersurfaces, lines and curves. The following three theorems are proved. Theorem~1. Let \(V\) be an infinite dimensional Banach space, \(X\subset P(V)\) a finite codimensional closed analytic subset of the projectivization \(P(V)\) of \(V\). If \(X\) itself is not a linear subspace of \(P(V)\), but \(X\) contains a finite codimensional linear subspace \(M\) of \(P(V)\), then \(X\) is singular, and its singular locus \(\text{Sing}(X)\) contains a closed finite codimensional analytic subset \(T\) of \(M\). Theorem~2. Let \(V,P(V)\) be as in theorem~1, \(A\subset V\) a finite dimensional linear subspace, and \(Y\) a closed analytic hypersurface of \(P(V)\). Then there is a linear subspace \(B\subset V\) such that \(A\subset B\), \(\dim(B)=\dim(A)+1\), and \(\text{Sing}(Y)\cap A =\text{Sing}(Y\cap B)\cap A\). Theorem~3. Let \(V\) and \(E\) be infinite dimensional complex Banach spaces, \(j:P(V)\to P(E)\) a closed embedding with \(j(P(V))\) a finite codimensional analytic subset of \(P(E)\). Then \(j\) is a linear isomorphism onto a finite codimensional closed linear subspace of \(P(E)\). The paper is well written and easy to follow.
reviewed by: Imre Patyi
Mathematics Subject Classification ID: 32K05
Mathematics Subject Classification ID: 58B12
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 2065403
zbMATH Keywords: infinite-dimensional projective space
zbMATH Keywords: Banach analytic set
zbMATH Keywords: Banach analytic manifold
zbMATH Keywords: singular Banach analytic set
zbMATH Keywords: Bertini theorem
author: E. Ballico
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2065403