Generators for the defining ideal of certain rational surfaces
type: Item
instance of: scholarly article
title: Generators for the defining ideal of certain rational surfaces
zbMATH Open document ID: 0731.14031
DOI: 10.1215/S0012-7094-91-06203-4
author: Alessandro Gimigliano
author: Anthony V. Geramita
published in: Duke Mathematical Journal
publication date: 1991-01-01T00:00:00Z
review text: One considers the rational surfaces \({\mathbb{P}}^ 2(Z)\) obtained by blowing-up the projective plane in the points of a finite set of points, Z. If \(E_ 1,...,E_ s\) are the (divisor classes on \({\mathbb{P}}^ 2(Z)\) of the) exceptional lines and \(E_ 0\) (the divisor class of) the proper transform of a line missing all the points of Z, then there is an integer \(\sigma\) which depends only on the Hilbert function of the ideal of Z in \({\mathbb{P}}^ 2\), such that \(tE_ 0-\sum E_ i\quad (t\geq \sigma)\) is very ample iff no \(\sigma\) elements of Z lie on a line of \({\mathbb{P}}^ 2\) [cf. \textit{E. D. Davis} and \textit{A. V. Geramita}, Math. Ann. 279, No.3, 435-448 (1988; Zbl 0657.14003)]. The main result of this paper says that for \(t\geq \sigma +1\) the ideals of the images by these embeddings are generated by quadrics.
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 4208269
zbMATH Keywords: syzygies
zbMATH Keywords: rational surfaces
zbMATH Keywords: blowing-up the projective plane
zbMATH Keywords: Hilbert function
zbMATH Keywords: quadrics
reviewed by: Nicolae Manolache
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4208269