Maximal and Cohen-Macaulay projective monomial curves
type: Item
instance of: scholarly article
title: Maximal and Cohen-Macaulay projective monomial curves
zbMATH Open document ID: 1111.14044
author: Les Reid
author: Leslie G. Roberts
published in: Journal of Algebra
publication date: 2007-01-26T00:00:00Z
review text: Let \( {\mathcal{S}}=\{a_1,\dots , a_k \}\) be a sequence of integers with \(0<a_1<\dots < a_k=d\) and \(\text{ gcd}( a_1,\dots ,a_k)=1\). Let \(R_{\mathcal{S} }= K[s^d, s^{d-a_1}t^{a_1}, s^{d-a_2}t^{a_2}, \dots ,t^d]\) be the homogeneous coordinate ring of a projective monomial curve over a field \(K\). Several authors [see e.g. \textit{S. Goto, N. Suzuki} and \textit{K. Watanabe}, Jap. J. Math., New. Ser. 2, 1--12 (1976; Zbl 0361.20066)] have given necessary and sufficient conditions for \(R_{\mathcal{S} }\) to be Cohen-Macaulay. In the present paper the authors continue the investigation of the Cohen-Macaulay property of the ring \(R_{\mathcal{S}}\) begun in [J. Algebra 291, No. 1, 171--186 (2005; Zbl 1085.14042)]. With this aim, the authors introduce the so-called maximal curves, observing that every Cohen-Macaulay projective monomial curve is maximal, but not conversely and that the test for maximality is simpler than for Cohen-Macaulayness. In particular they study closely the asymptotic properties of maximal and Cohen-Macaulay curves and show that the number of all curves of degree \(d\) that are Cohen-Macaulay grows exponentially in \(d\), but not as fast as the total number of curves and that maximal curves of degree \(d\) with sufficiently large embedding dimension relative to \(d\) are Cohen-Macaulay.
reviewed by: Caterina Cumino
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 5118826
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.04.014
OpenAlex ID: W2073806023
cites work: Problems and algorithms for affine semigroups
cites work: Q4134172
cites work: Hilbert functions of monomial curves
cites work: Non-Cohen–Macaulay Projective Monomial Curves with Positive h-Vector
cites work: Non-Cohen-Macaulay projective monomial curves.
cites work: Certain projective curves with unusual Hilbert function
DOI: 10.1016/J.JALGEBRA.2006.04.014
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description: scientific article; zbMATH DE number 5118826