The primitive length of a general \(k\)-gonal curve
type: Item
instance of: scholarly article
title: The primitive length of a general \(k\)-gonal curve
zbMATH Open document ID: 0845.14019
DOI: 10.1016/0019-3577(94)90022-1
author: Chang Ho Keem
author: Gerriet Martens
author: Marc Coppens
published in: Indagationes Mathematicae. New Series
publication date: 1995-09-04T00:00:00Z
review text: For a (smooth, irreducible projective) curve \(C\) of genus \(g \geq 4\), its Clifford sequence is defined as the set of Clifford indices \((d - 2r)\) for all nontrivial primitive linear series \(g^r_d\) (i.e. \(g^r_d\) is complete, and both it and its dual \(|K_C - g^r_d |\) verify that they are base-point free and are not the trivial \(g^0_0)\). The primitive length of \(C\) is defined to be the cardinality of the Clifford sequence. In Manuscr. Math. 77, No. 2/3, 237-264 (1992; Zbl 0786.14016), the authors began a detailed study of primitive linear series and Clifford lengths of curves. In the paper under review the authors continue this study, concentrating now on the general \(k\)-gonal curve and the Brill-Noether theory. More precisely, let \(3 < k < [(g + 3)/2]\) (the extreme cases where already studied in their previous paper), \(n \leq g\) and let \(\rho^1_n (g) = 2(n - 1) - g\) be the Brill-Noether number. Then the authors prove that there exist components of \(W^1_n\) different from \(W^1_k + W^0_{n - k}\), and that their dimension is given by \(\rho^1_n (g)\) (the problem of the uniqueness of such a component if \(\rho^1_n (g) > 0\) remains open). They also give some evidence for the following conjecture: Any number in the Clifford sequence that is smaller than \([(g - 1)/2]\) is the Clifford index of a multiple of the \(g^1_k\).
reviewed by: Enrique Arrondo
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 639722
zbMATH Keywords: \(k\)-gonal curve
zbMATH Keywords: Clifford sequence
zbMATH Keywords: linear series
zbMATH Keywords: primitive length
zbMATH Keywords: Clifford lengths of curves
zbMATH Keywords: Brill-Noether theory
MaRDI profile type: Publication
full work available at URL: 0019-3577(94)90022-1
OpenAlex ID: W2051471831
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description: scientific article; zbMATH DE number 639722