On the projective normality of Enriques surfaces (with an appendix by Angelo Felice Lopez and Alessandro Verra)
type: Item
instance of: scholarly article
title: On the projective normality of Enriques surfaces (with an appendix by Angelo Felice Lopez and Alessandro Verra)
zbMATH Open document ID: 1081.14052
author: Luis Giraldo
author: Angelo Felice Lopez
author: Roberto Muñoz
published in: Mathematische Annalen
publication date: 2002-12-01T00:00:00Z
review text: The question of projective normality of linearly normal curves of fixed genus \(g\) and degree at least \(2g +1\) is well known and has been studied by several authors. For surfaces some results have been proved in this direction as for example \textit{L. Ein} and \textit{R. Lazarsfeld} [Invent. Math. 111, 51--67 (1993; Zbl 0814.14040)] for varieties of any dimension and more precise results on syzygies or projective normality of surfaces, as it is the case for example by \textit{D. C. Butler} [J. Differ. Geom. 39, 1--34 (1994; Zbl 0808.14024)] for ruled surfaces, \textit{Y. Homma} [Nat. Sci. Rep. Ochanomizu Univ. 31, 61--73 (1980; Zbl 0486.14001)] for the case of elliptic ruled surfaces, \textit{F. Gallego} and \textit{B. P. Purnaprajna} [J. Algebra 186, 597--625 (1996; Zbl 0948.14029)] on projective normality and syzigies of elliptic ruled surfaces, surfaces of general type and Enriques surfaces, among other authors. As the authors of this paper note for the case of \(K3\) surfaces any linearly normal \(K3\) surface is projectively normal and its ideal is generated by quadrics and cubics. They pose the same question for Enriques surfaces and prove in this note their main result, which is stated as follows: Theorem (1.1). Let \(S \subset \mathbb P^{g-1}\) be a linearly normal smooth irreducible Enriques surface. If \(g=6\) and \({\mathcal O}_S(1)\) gives an embedding in a quadric then \(S\) is \(j\)-normal for \(j\geq 3\) and its homogeneous ideal is generated by quadrics and cubics. If either \(g \geq 7\) or \(g=6\) and \({\mathcal O}_S(1)\) does not give an embedding in a quadric then \(S\) is 3-regular in the sense of Castelnuovo-Mumford. In particular \(S\) is projectively normal and its homogeneous ideal is generated by quadrics and cubics. In theorem 1.4 the authors study the geometry of a curve \(C\) on a smooth Enriques surface, whenever \(C\) has gonality \(k \geq 3\). In corollary 1.10 they show that if \(g\) is the genus of \(C\) and \(e\) the Clifford i…
reviewed by: Isidro Nieto Baños
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 1837350
MaRDI profile type: Publication
Wikidata QID: Q60143215
full work available at URL: s00208-002-0333-5
OpenAlex ID: W2088864958
DOI: 10.1007/S00208-002-0333-5
description: scientific article; zbMATH DE number 1837350