Postulation of canonical curves in \({\mathbb{P}}^ 3\)
type: Item
instance of: scholarly article
title: Postulation of canonical curves in \({\mathbb{P}}^ 3\)
zbMATH Open document ID: 0576.14033
DOI: 10.1007/BF01458014
author: Mei-Chu Chang
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164084
review text: A non-singular curve \(Y\subset {\mathbb{P}}^ 3\) of genus g is said to be ''canonical'' if its plane section is a canonical divisor, i.e. if the canonical bundle \(K_ Y\simeq {\mathcal O}_ Y(1)\). In other words, \(Y\subset {\mathbb{P}}^ 3\) is the projection of its canonical image in \({\mathbb{P}}^{g-1}\). Thus the canonical curves of given genus form a single irreducible family. Now recall that a curve \(Y\subset {\mathbb{P}}^ 3\) is said to have ''maximal rank'' if the restriction maps: \(H^ 0({\mathcal O}_{{\mathbb{P}}^ 3}(n))\to H^ 0({\mathcal O}_ Y(n))\) have maximal rank for all n. Given any family of curves in \({\mathbb{P}}^ 3\), it is a natural question whether its general member has maximal rank. While one usually expects the answer to be ''yes'', the case of canonical curves is somewhat intriguing in that in the first two nontrivial cases, namely those of genus 5 and 6, the answer turns out to be ''NO'' [proofs were given by \textit{L. Gruson} and \textit{C. Peskine} in Algebr. Geom., Proc., Tromsø Symp. 1977, Lect. Notes Math. 687, 31-59 (1978; Zbl 0412.14011) and by \textit{R. Hartshorne} and \textit{I. Sols} in J. Reine Angew. Math. 325, 145-152 (1981; Zbl 0448.14004)]. Recently the question was taken up by \textit{E. Ballico} [Ark. Mat. 22, 139-151 (1984; Zbl 0546.14018)], who showed that a general canonical curve of genus \(g=7, 8, 9\) or 11 has maximal rank. In this note we prove the following theorem: A general canonical curve of genus \(g\geq 7\) in \({\mathbb{P}}^ 3\) has maximal rank.
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 3920660
zbMATH Keywords: postulation
zbMATH Keywords: general canonical curve
zbMATH Keywords: genus
zbMATH Keywords: maximal rank
MaRDI profile type: Publication
cites work: On the postulation of canonical curves in \(\mathbb P^3\)
cites work: Genre des courbes de l'espace projectif
cites work: Q3894039
OpenAlex ID: W2051825082
description: scientific article; zbMATH DE number 3920660