On the projective normality of complete linear series on an algebraic curve
type: Item
instance of: scholarly article
title: On the projective normality of complete linear series on an algebraic curve
zbMATH Open document ID: 0594.14010
DOI: 10.1007/BF01388754
author: Robert Lazarsfeld
author: Mark Green
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143305
review text: Main result of this beautiful paper: Let L be a very ample line bundle on a smooth curve X of genus g, \(h^ 0(L)=r+1\), \(\deg (L)=2g+1-k\) with \(2k+1\leq g\) and let \(Y\subset {\mathbb{P}}^ r\) be the embedding of X determined by L; \((a)\quad L\) is not normally generated if and only if there is n, \(1\leq n\leq r-2\), and an effective divisor D on Y, \(\deg (D)\geq 2n+2\), such that \(H^ 1(Y,L^ 2(-D))=0\) and D spans an n-plane in which D fails to impose independent conditions on quadrics; \((b)\quad if\) \(\deg (L)\geq 2g+1-2h^ 1(L)-Cliff(X)\), then L is normally generated. Furthermore if \(g>>Cliff(X)\), the authors classify the line bundles with \(\deg (L)=2g-2h^ 1(L)-Cliff(X)\) and not normally generated. They discuss several conjectures on syzygies of projective curves and the Clifford index of curves. [Conjecture 3.8 is now proved by the reviewer; see Proc. Am. Math. Soc. 97, 217-218 (1986; Zbl 0591.14020).]
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 3955091
zbMATH Keywords: normally generated line bundle
zbMATH Keywords: syzygy
zbMATH Keywords: genus
zbMATH Keywords: gonality
zbMATH Keywords: Clifford index of curves
MaRDI profile type: Publication
cites work: Q3347989
cites work: Petri's approach to the study of the ideal associated to a special divisor
cites work: Q4123438
cites work: Q3324080
cites work: Koszul cohomology and the geometry of projective varieties. Appendix: The nonvanishing of certain Koszul cohomology groups (by Mark Green and Robert Lazarsfeld)
cites work: On projective normality and defining equations of a projective curve of genus three embedded by a complete linear system
cites work: On the equations defining a projective curve embedded by a non-special divisor
cites work: Q5185016
cites work: Normal generation of line bundles of degree 2p-2 on curves
cites work: Funktionen von vorgegebener Ordnung auf komplexen Kurven.
cites work: Über den Clifford-Index algebraischer Kurven.
cites work: Varieties of special divisors on a curve. II.
cites work: Symmetric Products and Jacobians
cites work: Lectures on Curves on an Algebraic Surface. (AM-59)
cites work: On Self-Intersection Number of a Section on a Ruled Surface
cites work: On Petri's analysis of the linear system of quadrics through a canonical curve
OpenAlex ID: W1998446953
description: scientific article; zbMATH DE number 3955091