Determinant surfaces of rank 2 bundles on \(\mathbb{P}^ 3\)
type: Item
instance of: scholarly article
title: Determinant surfaces of rank 2 bundles on \(\mathbb{P}^ 3\)
zbMATH Open document ID: 0834.14017
author: Shigeharu Takayama
published in: Tokyo Journal of Mathematics
publication date: 1996-04-01T00:00:00Z
review text: The aim of this paper is to study the relationship between stable vector bundles of rank two on projective three space and their determinant surface determined by two sections of the bundle. A general surface has Picard number \(\rho (S) = 1\). A smooth surface \(S\) is not general because its Picard number is at least two by the following result of the author: A smooth surface \(S \subset \mathbb{P}^3\) occurs as a determinant of a rank two vector bundle \({\mathcal E}\) on \(\mathbb{P}^3\) if and only if \(S\) has a surjective morphism onto \(\mathbb{P}^1\). In the following the author estimates \(\rho (S)\). There is an estimate of \(\rho (S)\) from below in terms of the behaviour of \({\mathcal E}\) under the restriction to lines and planes. In particular, defining jumping planes there is a sufficient condition for \(S\) to have Picard number \(\rho (S) \geq 3\).
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 14J70
zbMATH DE Number: 737433
zbMATH Keywords: stable vector bundles of rank two
zbMATH Keywords: determinant surface
zbMATH Keywords: Picard number
reviewed by: Peter Schenzel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270127955
OpenAlex ID: W2072574631
DOI: 10.3836/TJM/1270127955
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description: scientific article; zbMATH DE number 737433