A remark on the syzygies of the generic canonical curves
type: Item
instance of: scholarly article
title: A remark on the syzygies of the generic canonical curves
zbMATH Open document ID: 0632.14024
author: Lawrence Ein
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: If C is a genus g nonhyperelliptic curve, then it is defined the canonical ring R as \(\oplus_{n}H^ 0(\omega^ n_ C)\quad which\) is a graded module over the polynomial ring \(S=Symm(H^ 0(\omega_ C))\). Then it is defined a minimal graded Koszul resolution of \(S/(irrelevant\quad ideal)\) and therefore the Koszul cohomology group \(K_{pq}(R)=K_{pq}(C)\). The paper, extending results of \textit{M. Green} and \textit{R. Lazarsfeld} [J. Differ. Geom. 19, 168-171 (1984; Zbl 0559.14008)] and using results of Bayer and Stillman based on the computer program Macaulay \(proves:\) K\({}_{p2}(X)=0\), when X is a curve of genus n, implies that \(K_{p2}(C)=0\) for a generic curve of genus \(m,\) provided that \(m=n mod(p+1)\) and \(m>n-1;\) K\({}_{22}(C)=0\) for \(g>6\) and \(K_{32}(C)=0\) for \(g>8\).
reviewed by: Paolo Valabrega
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 14-04
Mathematics Subject Classification ID: 13D25
zbMATH DE Number: 4027657
zbMATH Keywords: syzygies
zbMATH Keywords: nonhyperelliptic curve
zbMATH Keywords: canonical ring
zbMATH Keywords: Koszul cohomology group
zbMATH Keywords: computer program Macaulay
MaRDI profile type: Publication
Wikidata QID: Q115184398
full work available at URL: https://doi.org/10.4310/jdg/1214441376
OpenAlex ID: W1596559456
DOI: 10.4310/JDG/1214441376
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description: scientific article; zbMATH DE number 4027657