On minimal free resolutions of projective varieties of degree = codimension + 2
type: Item
instance of: scholarly article
title: On minimal free resolutions of projective varieties of degree = codimension + 2
zbMATH Open document ID: 0803.13007
DOI: 10.1016/0022-4049(93)90112-7
author: Le Tuan Hoa
published in: Journal of Pure and Applied Algebra
publication date: 1993-11-09T00:00:00Z
review text: The paper is a supplement to the joint work of the author, \textit{J. Stückrad} and \textit{W. Vogel} [see J. Pure Appl. Algebra 71, No. 2/3, 203-231 (1991; Zbl 0749.14032)] on the structure of non-degenerate, reduced and irreducible varieties \(X \subset \mathbb{P}^ n_ k\), \(k\) an algebraically closed field, of `almost minimal' degree. In general one has \(\deg X\geq\) codim \(X+1\), while `almost minimal' means \(\deg X = \text{codim} X + 2\). In the present paper there is a discussion of the minimal free resolution of \(X\) when it is of `almost minimal' degree. In the case \(X\) is arithmetically Cohen-Macaulay the coordinate ring is an extremal Gorenstein ring of size 2 [see the reviewer, J. Algebra 64, 93-101 (1980; Zbl 0449.13008)]. So it has a linear resolution and there are explicit formulas for the Betti numbers. In the general case the author uses the 2-regularity of \(X\) in order to show that the minimal generators of the syzygy modules are concentrated in at most two degrees. This implies several results about the Betti numbers of \(X\), in particular in large homological degrees. These statements become more explicit in small dimensions (for curves) or under additional assumptions.
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13C40
zbMATH DE Number: 427766
zbMATH Keywords: variety of almost minimal degree
zbMATH Keywords: minimal free resolution
zbMATH Keywords: arithmetically Cohen-Macaulay
zbMATH Keywords: Betti numbers
zbMATH Keywords: minimal generators of the syzygy modules
MaRDI profile type: Publication
full work available at URL: 0022-4049(93)90112-7
OpenAlex ID: W1983073268
cites work: Q3199546
cites work: Linear free resolutions and minimal multiplicity
cites work: Classification of projective varieties of \(\Delta\)-genus one
cites work: Residues and zero-cycles on algebraic varieties
cites work: Towards a structure theory for projective varieties of degree = codimension + 2
cites work: Lectures on Curves on an Algebraic Surface. (AM-59)
cites work: Castelnuovo's regularity of graded rings and modules
cites work: Über die freien Auflösungen extremaler Cohen-Macaulay-Ringe
cites work: Q3744271
description: scientific article; zbMATH DE number 427766