Surfaces of sectional genus \(\leq 8\) with no trisecant lines
type: Item
instance of: scholarly article
title: Surfaces of sectional genus \(\leq 8\) with no trisecant lines
zbMATH Open document ID: 0793.14027
DOI: 10.1007/BF01194243
author: Marco Andreatta
published in: Archiv der Mathematik
publication date: 1994-08-25T00:00:00Z
review text: Let \(S\) be a smooth projective surface over \(\mathbb{C}\) and \(L\) a line bundle on \(S\). According to a paper by \textit{M. Beltrametti} and \textit{A. J. Sommese} [``Zero cycles and \(k\)-th order embeddings of smooth projective surfaces'', Proc. Projective Surfaces and their Classification, Cortona (1988), Symp. Math., INDAM] we say that \(L\) is \(k\)-very ample, for \(k \geq 2\), if for any 0-dimensional subscheme \((Z,{\mathcal O}_ Z)\) of length \(k+1\) the restriction map \(\Gamma (L) \to \Gamma (L \bigotimes {\mathcal O}_ Z)\) is a surjection. For \(k=1\) this means that \(L\) is very ample; for \(k=2\) on the other hand it means that if \(S \subset \mathbb{P}^ N\) is the embedding defined by the global sections of \(L\) (i.e. \(L={\mathcal O} (1)_{| S})\) then there are no 0-dimensional subscheme of length 3 on a line; in this case we will say that \(S \subseteq \mathbb{P}^ N\) has no 3-secant lines. A classical problem, if \(L\) is very ample, is to give a classification of the pairs \((S, L) \) with low numerical invariants, for instance with low sectional genus. -- In this paper we want to classify the pair \((S,L)\) with sectional genus \(\leq 8\) and such that \(L\) is \(k\)-very ample for \(k \geq 2\). For \(g \leq 5\) the classification was carried out in the paper by \textit{M. Beltrametti} and \textit{A. J. Sommese} in Algebraic Geometry, Proc. Int. Conf., L'Asquila 1988, Lect. Notes Math. 1417, 24-51 (1990; Zbl 0706.14007) and for \(g=6\) by \textit{A. Lanteri} [``2-spanned surfaces of sectional genus six'', Ann. Mat. Pura Appl., IV. Ser. 165, 197-216 (1993)]. We therefore consider the cases \(g=7\) and 8.
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 571900
zbMATH Keywords: classification of \(k\)-very ample line bundles
zbMATH Keywords: 3-secant lines
MaRDI profile type: Publication
cites work: On the 2 very ampleness of the adjoint bundle. (With an appendix by E. Ballico)
cites work: A Characterization of the Veronese Surface
cites work: On Reider's method and higher order embeddings
cites work: Q3487534
cites work: Q3919817
cites work: 2-spanned surfaces of sectional genus six
cites work: Formules pour les trisécantes des surfaces algébriques. (Formulas for the trisecants of algebraic surfaces)
cites work: Classification of algebraic surfaces with sectional genus less than or equal to six. I: Rational surfaces
cites work: Curvilinear enumerative geometry
cites work: Hyperplane sections of projective surfaces. I: The adjunction mapping
full work available at URL: https://doi.org/10.1007/bf01194243
OpenAlex ID: W2067030786
Recommended article: On surfaces in \(\mathbb{P}^{6}\) with no trisecant lines
Recommended article: The non-existence of a smooth sectionally non-special surface of degree 11 and sectional genus 8 in the projective fourspace
Recommended article: Surfaces in \(\mathbb{P}^4\) with no quadrisecant lines
Recommended article: Q3784249
Recommended article: Surfaces in \(\mathbb{P}^5\) which do not admit trisecants
Recommended article: On the genus of a hyperplane section of a geometrically ruled surface
Recommended article: Every Curve of Genus not Greater Than Eight Lies on a K3 Surface
Recommended article: Classification of algebraic non-ruled surfaces with sectional genus less than or equal to six
Recommended article: Surfaces with a hyperelliptic hyperplane section
description: scientific article; zbMATH DE number 571900