An algorithm for computing minimal curves
type: Item
instance of: scholarly article
title: An algorithm for computing minimal curves
zbMATH Open document ID: 0896.14013
author: Silvia Guarrera
author: Alessandro Logar
author: Emilia Mezzetti
published in: Archiv der Mathematik
publication date: 1997-07-06T00:00:00Z
review text: Let \(\mathbb{P}^3\) be the 3-dimensional projective space over an algebraically closed field \(K\) of characteristic 0, and \(R\) be the graded polynomial ring \(K[x_0,x_1,x_2,x_3]\). A curve \(C\) of \(\mathbb{P}^3\) will be a closed subscheme of \(\mathbb{P}^3\), purely of dimension 1, locally Cohen-Macaulay, that is without embedded points, defined by an ideal sheaf \({\mathcal I}_C\). Two curves \(C\) and \(C'\) of \(\mathbb{P}^3\) are said to be geometrically linked if \(C\) and \(C'\) do not have common irreducible components and their union is a complete intersection. The equivalence relation generated by this relation is called liaison. Hence \(C\) and \(C'\) belong to the same liaison class if there exist curves \(C_0= C,C_1,\dots, C_n= C'\) such that \(C_1\) and \(C_{i+1}\) are geometrically linked, for \(i=0,\dots, n-1\). If in particular the integer \(n\) is even, then we say that \(C\) and \(C'\) belong to the same biliaison class. Let \(M_C:= \bigoplus_{n\in\mathbb{Z}} H^1{\mathcal I}_C(n)\) be the Hartshorne-Rao module of the curve \(C\): it is a graded \(R\)-module of finite length which, up to a shift in degrees, characterizes the biliaison class of \(C\). \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184-185 (1990; Zbl 0717.14017)] have proved that in every biliaison class of non arithmetically Cohen-Macaulay curves there exists a minimal curve, unique up to a deformation with constant cohomology and Hartshorne-Rao module, that is a curve \(C\) such that, if \(C_1\) is any curve in the biliaison class of \(C\), then \(M_{C_1}= M_C(-n)\), with \(n\geq 0\). They have moreover given an algorithm for the computation of a minimal curve, based on the computation and the analysis of the minors of given orders of some submatrices of the second syzygy matrix \(\sigma_2\) of \(M\). The aim of this paper is to give an improvement to the algorithm for computing a minimal curve of a given Hartsho…
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14M06
Mathematics Subject Classification ID: 14Q05
zbMATH DE Number: 1028197
zbMATH Keywords: linkage
zbMATH Keywords: liaison
zbMATH Keywords: biliaison class
zbMATH Keywords: Hartshorne-Rao module
describes a project that uses: CoCoA
describes a project that uses: Macaulay2
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: s000130050058
OpenAlex ID: W2095051675
cites work: Some features of \(CoCoA 3\)
cites work: Q4002797
cites work: Q4376923
cites work: Q3997640
cites work: Q4764201
cites work: Liaison among curves in \(\mathbb{P}^3\)
cites work: Generalized subresultants for computing the Smith normal form of polynomial matrices
DOI: 10.1007/S000130050058
description: scientific article; zbMATH DE number 1028197