Decomposition of normal bundle of smooth surfaces of \(\mathbb{P}_ 4\) and double structures on solids in \(\mathbb{P}_ 5\)
type: Item
instance of: scholarly article
title: Decomposition of normal bundle of smooth surfaces of \(\mathbb{P}_ 4\) and double structures on solids in \(\mathbb{P}_ 5\)
zbMATH Open document ID: 0798.14028
DOI: 10.1215/S0012-7094-93-06904-9
author: Bénédicte Basili
author: Christian Peskine
published in: Duke Mathematical Journal
publication date: 1994-11-03T00:00:00Z
review text: This paper deals with the problem of characterizing complete intersections among smooth codimension two subvarieties in projective space. The first characterization is given by means of the splitting of the normal bundle. The authors prove this result for surfaces (so that this characterization holds for dimension at least two, being clearly false for curves). The key point is to prove that the summands of the normal bundle depend numerically on the hyperplane section, and the result is then obtained by restricting to a general hyperplane section. One could also try to characterize complete intersections by giving the normal bundle as an extension of two line bundles. This holds for dimension at least four, but it is not true for surfaces (although the authors prove it when the line bundles depend numerically on the hyperplane section). In the case of a threefold \(X\), the authors consider a natural double structure on \(X\), which appears from the given extension and is the zero section of a rank two vector bundle on \(\mathbb{P}^ 5\). Then they prove that \(X\) is a complete intersection if and only if this bundle splits.
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14M07
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 179230
zbMATH Keywords: characterizing complete intersections
zbMATH Keywords: codimension two subvarieties
zbMATH Keywords: splitting of the normal bundle
zbMATH Keywords: threefold
zbMATH Keywords: double structure
reviewed by: Enrique Arrondo
MaRDI profile type: Publication
cites work: Transplanting Cohomology Classes in Complex-Projective Space
cites work: On the normal bundle of curves on smooth projective surfaces
cites work: Q3331324
cites work: On the topology of complex projective manifolds
cites work: Q3972823
full work available at URL: s0012-7094-93-06904-9
OpenAlex ID: W1573667140
description: scientific article; zbMATH DE number 179230