Liaison and Cohen-Macaulayness conditions
type: Item
instance of: scholarly article
title: Liaison and Cohen-Macaulayness conditions
zbMATH Open document ID: 1221.14054
published in: Collectanea Mathematica
publication date: 2011-06-22T00:00:00Z
review text: Let \(K\) be an algebraically closed field, \(\mathbb{P}_{K}^{n}\) the projective space of dimension \(n\) over \(K\), \(C\) a projective curve in \( \)\(\mathbb{P}_{K}^{n}\), i.e. \(C\) is an equidimensional projective subscheme of dimension 1, hence a locally Cohen- Macaulay projective subscheme of dimension 1. There are positive integers \(\beta_{1}\leq\dots \leq\beta_{n-1}\) and a complete intersection \(Y\) of type \((\beta_{1},\dots ,\beta_{n-1})\) such that \(Y\) contains \(C\). Let \(C'\) be the curve algebraically linked to \(C\) by \(Y\) and \(\rho C\) be the regularity of the Hilbert function of \(C\). By the work of \textit{F. Cioffi, M. G. Marinari} and \textit{L. Ramella} [Collect. Math. 60, No. 1, 89--100 (2009; Zbl 1188.14020)], it is proved that if \(C\) is a space curve such that \(\text{reg}(C)=\text{reg}(Y)-1>\rho C+1\) then \(C\) is a plane curve, hence arithmetically Cohen- Macaulay, here \(\text{reg}(*)\) denotes the Castelnuovo-Mumford regularity. The authors show \(\text{reg}(Y)-\text{reg}(C)\geq0\) when \(\text{reg}(C)>\rho C+1\) and give some necessary or sufficient conditions for \(C'\) being connected by means of the regularity of \(C\) (Theorem 3.1, Proposition 3.6) By the work of \textit{E. D. Davis} [in: Curves Semin. Queen's, Vol. VII, Queen's Pap. Pure Appl. Math. 85, Exposé F, 14 p. (1990; Zbl 0736.14011)], when \(\text{deg}(Y)-\text{deg}(C)\leq5\), a smooth integral space curve \(C\) with the same postulation as an arithmetically normal curve is necessarily arithmetically Cohen-Macaulay. Here \(\text{deg}(Y)-\text{deg}(C)\) measures the difference between \(C\) and the complete intersection \(Y\) in term of degree of curves. The authors prove the analogous result of Davis with more general hypotheses that \(C\) is any space curve with \(\text{reg}C>\rho C+1\) and \(\text{deg}(Y)-\text{deg}(C)\leq5\) (Theorem 4.2). They further show the result holds for non-degenerate equidimensional and locally Cohen-Macaulay subschemes \(X\) of c…
reviewed by: Kuei-Nuan Lin
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14M06
Mathematics Subject Classification ID: 14M10
zbMATH DE Number: 5911552
zbMATH Keywords: Cohen-Macaulayness
zbMATH Keywords: Liaison
zbMATH Keywords: Hilbert function
zbMATH Keywords: Castelnuovo-Mumford regularity
author: Francesca Cioffi
author: Roberta Di Gennaro
describes a project that uses: CoCoA
MaRDI profile type: Publication
full work available at URL: s13348-010-0010-z
OpenAlex ID: W1963871643
cites work: Two-codimensional buchsbaum subschemes of pnvia their hyperplane sections
cites work: Bounding cohomological Hilbert functions by hyperplane sections
cites work: Regularity bounds by minimal generators and Hilbert function
cites work: Q3348001
cites work: Gorenstein Algebras and the Cayley-Bacharach Theorem
cites work: The equations of space curves on a quadric
cites work: Genre des courbes de l'espace projectif
cites work: Q4143433
cites work: Q4887547
cites work: Curves in the double plane
cites work: Q4202167
cites work: Q3676243
cites work: Degrees of generators of ideals defining curves in projective space
cites work: The Hilbert function of generic plane sections of curves of \({\mathbb P}^ 3\)
cites work: Introduction to liaison theory and deficiency modules
cites work: Q5288193
cites work: Integral subschemes of codimension two
cites work: Q3973283
cites work: Q5710779
cites work: Q4251934
DOI: 10.1007/S13348-010-0010-Z
description: scientific article; zbMATH DE number 5911552