Minimal free resolutions of monomial curves in \({\mathbb{P}}^ 3_ k\)
type: Item
instance of: scholarly article
title: Minimal free resolutions of monomial curves in \({\mathbb{P}}^ 3_ k\)
zbMATH Open document ID: 0542.14022
DOI: 10.1016/0024-3795(84)90163-0
author: H. Bresinsky
published in: Linear Algebra and its Applications
publication date: 1984-01-01T00:00:00Z
review text: Let \(0=n_ 0<n_ 1<n_ 2<n_ 3\) be integers with \((n_ 1,n_ 2,n_ 3)=1.\) Let P be the prime ideal of the monomial curve with parametric representation \(x_ i=t_ 0^{n_ B-n_ i}t_ 1^{n_ i}, 0\leq i\leq 3,\) in \({\mathbb{P}}^ 3_ k\) with \(\mu(P)=\mu\) the smallest number of generators of P. Let \(R=k[x_ 0,...,x_ 3]\). It is shown that \(0\to R^{\mu -3}\to^{\phi_ 3}R^{2\mu -4}\to^{\phi_ 2}R^{\mu}\to^{\phi_ 1}R\to R/P\to 0\) is a minimal finite free resolution for \(\mu \geq 3\). The mappings \(\phi_ i\), 1\(\leq i\leq 3\), are determined explicitly.
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 3861327
zbMATH Keywords: Buchsbaum-Eisenbud criterion
zbMATH Keywords: prime ideal of the monomial curve
zbMATH Keywords: minimal finite free resolution
MaRDI profile type: Publication
full work available at URL: 0024-3795(84)90163-0
OpenAlex ID: W2007334861
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description: scientific article; zbMATH DE number 3861327