Classification of toric curves in projective space, Rao module and liaison
type: Item
instance of: scholarly article
title: Classification of toric curves in projective space, Rao module and liaison
zbMATH Open document ID: 0937.14037
author: Laurence Coudurier
author: Marcel Morales
published in: Journal of Algebra
publication date: 1999-09-09T00:00:00Z
review text: A toric curve \(C\) in \({\mathbb P}^3\) is given by a parameterization \(x = s^a\), \(y = s^{a-c}t^c\), \(z = s^{a-b}t^b\), \(w = t^a\) with \(a,b,c \in {\mathbb N}\) relatively prime positive integers. If \(a>b>c>0\) then \(C\) is called a \textit{monomial curve}. The paper addresses the following question. Given a toric curve \(C\) in \({\mathbb P}^3\), is there another toric curve \(C'\) linked to \(C\) in an even (resp. odd) number of steps? For the case of monomial curves and linking in an even number of steps (i.e.\ the case of even liaison classes of monomial curves), this was answered by \textit{H. Bresinsky} and \textit{C. Huneke} [J. Reine Angew. Math. 365, 33-66 (1986; Zbl 0574.14029)]. Using an explicit minimal generating set of the ideal \(I(C)\) due to the second author, the authors find the minimal free resolution of \(I(C)\) (including the maps). This allows them to compute the Hartshorne-Rao module \(M(C)\) and conclude about the liaison class. In particular they prove a conjecture of Bresinsky and Huneke [op. cit.] that if \(C\) is a monomial curve such the number of minimal generators of \(I(C)\) is at least 5 then there is no monomial curve linked to \(C\) in an odd number of steps.
reviewed by: Juan C. Migliore
Mathematics Subject Classification ID: 14M06
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 1271336
zbMATH Keywords: liaison
zbMATH Keywords: toric curve
zbMATH Keywords: monomial curve
zbMATH Keywords: Hartshorne-Rao module
zbMATH Keywords: syzygy
zbMATH Keywords: oddly linked curves
zbMATH Keywords: linkage
MaRDI profile type: Publication
full work available at URL: jabr.1998.7739
OpenAlex ID: W2076008553
cites work: What makes a complex exact?
cites work: Minimal free resolutions of monomial curves in \({\mathbb{P}}^ 3_ k\)
cites work: Liasion of monomial curves in P3
cites work: On the canonical module for monomial curves
cites work: Q3997640
cites work: Equations of monomial varieties in codimension two
cites work: Liaison des variétés algébriques. I
cites work: Liaison among curves in \(\mathbb{P}^3\)
cites work: Q3744271
DOI: 10.1006/JABR.1998.7739
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description: scientific article; zbMATH DE number 1271336