Embeddings of blown-up plane
type: Item
instance of: scholarly article
title: Embeddings of blown-up plane
zbMATH Open document ID: 0943.14003
DOI: 10.1023/A:1005106123696
published in: Geometriae Dedicata
publication date: 2000-02-28T00:00:00Z
review text: Let \(X\) be a smooth projective surface defined over the complex number field \(\mathbb{C}\). Then there is the following problem which is very interesting: Problem: If \(X\) is a blown up surface of a smooth surface \(S\), then study the property of the pullback bundle \(L\) of a line bundle \(A\) on \(S\). \textit{G. Xu} [Manuscr. Math. 86, 195-197 (1995; Zbl 0836.14004) and J. Reine Angew. Math. 469, 199-209 (1995; Zbl 0833.14028)], \textit{O. Küchle} [Math. Ann. 304, No. 1, 151-155 (1996; Zbl 0834.14024)], and \textit{K. Yokoyama} [J. Math. Soc. Japan 51, No. 3, 523-533 (1999)] have given some criteria for the pullback bundle of ample line bundle to be ample. \textit{J. D'Almeida} and \textit{A. Hirschowitz} [Math. Z. 211, No. 3, 479-483 (1992; Zbl 0759.14004)] considered the case of points in generic position on \(\mathbb{P}^{2}\), and gave a criterion for the pullback bundle to be very ample. In this paper, the author considers the above problem in the case in which \(S\) is a projective space \(\mathbb{P}^{2}\) and gives a criterion for the pullback to be \(k\)-very ample. Here the notion of \(k\)-very ampleness of a line bundle \(L\) is defined by the following: For any \(0\)-dimensional subscheme \(Z\) of length \(k+1\) on \(S\), the map \(H^{0}(L)\to H^{0}(L_{Z})\) is surjective. By definition \(0\)-very-ampleness (resp. \(1\)-very ampleness) is equivalent to spannedness (resp. very ampleness). So the criterion in this paper is a generalization of the paper by D'Almeida and Hirschowitz cited above. Though the criterion in this paper is true, there are many mistakes and you need to read this paper carefully.
reviewed by: Yoshiaki Fukuma
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14E25
zbMATH DE Number: 1335049
zbMATH Keywords: embeddings
zbMATH Keywords: \(k\)-very ampleness
zbMATH Keywords: blown-up plane
zbMATH Keywords: pullback bundle
author: Halszka Tutaj-Gasińska
MaRDI profile type: Publication
full work available at URL: a:1005106123696
OpenAlex ID: W102586141
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description: scientific article; zbMATH DE number 1335049