Very ample line bundles on regular surfaces obtained by projection
type: Item
instance of: scholarly article
title: Very ample line bundles on regular surfaces obtained by projection
zbMATH Open document ID: 1108.14008
author: Atsushi Noma
published in: Journal of Algebra
publication date: 2007-01-11T00:00:00Z
review text: Let \(X \subset \mathbb P^N\) be a smooth projective algebraic surface defined over an algebraically closed field, with hyperplane bundle \(\mathcal L = \mathcal O_X(1)\), degree \(d\), and sectional genus \(g\). Let \(\sigma: \widehat{X} \to X\) be the blowing-up at distinct points \(x_1, \dots , x_m \in X\) and consider the line bundle \(\widehat{\mathcal L}:= \sigma^* \mathcal L \otimes \mathcal O_{\widehat{X}} (-E_1 - \dots -E_m)\), where \(E_i\) is the exceptional divisor corresponding to \(x_i\). The problem of finding conditions for the very ampleness of \(\widehat{\mathcal L}\) in terms of the configuration of points \(x_1, \dots , x_m\) has been considered by many authors, e. g., \textit{E. M. Bese} [Math. Ann. 262, 225--238 (1983; Zbl 0491.14009)], \textit{F. Catanese} and \textit{M. Franciosi} [Proc. Hirzebruch 65 Conf. Algebraic Geometry, Ramat Gan, 1993, Isr. Math. Conf. Proc. 9, 109--140 (1996; Zbl 0855.14004)], \textit{M. Coppens} [Pac. J. Math. 202, No. 2, 313--327 (2002; Zbl 1059.14011)]. In the paper under review the author proves the following. Suppose that \(h^1(X, \mathcal O_X)=0\) and \(m \leq d-2g-1\). Then \(\widehat{\mathcal L}\) is very ample if and only if for all \(l\), \(1 \leq l \leq m\), any \(l\) distinct points of \(\{x_1, \dots ,x_m\}\) do not lie on any rational normal curve of degree \(l\) contained in \(X\). A key role in the proof is played by inner projections of \(X\) from a subset of \(\{x_1, \dots ,x_m\}\). As an application, the author determines precise conditions ensuring the very ampleness of some line bundles arising in the classification of surfaces of sectional genus \(2\) and \(3\), namely rational conic fibrations (\(g=2\) and \(3\)) and Bordiga surfaces (\(g=3\)).
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14E25
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 5115364
zbMATH Keywords: very ample line bundle
zbMATH Keywords: regular (projective) surface
zbMATH Keywords: linear projection
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.05.014
OpenAlex ID: W2042320542
cites work: Q4750781
cites work: The adjunction theory of complex projective varieties
cites work: On the spannedness and very ampleness of certain line bundles on the blow-ups of \(P^ 2 \)and \(F^ r\).
cites work: Q4860496
cites work: Very ample linear systems on blowings-up at general points of smooth projective varieties.
cites work: Some non special projective embeddings of rational surfaces
cites work: Birational morphisms to \({\mathbb{P}}^ 2:\) An ideal-theoretic perspective
cites work: Joins and Intersections
cites work: Q4143433
cites work: Q3329577
cites work: Smooth projective varieties with extremal or next to extremal curvilinear secant subspaces
cites work: On the existence of some surfaces
cites work: Q3270271
DOI: 10.1016/J.JALGEBRA.2006.05.014
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description: scientific article; zbMATH DE number 5115364