Degree bounds for the defining equations of arithmetically Cohen-Macalay varieties
type: Item
instance of: scholarly article
title: Degree bounds for the defining equations of arithmetically Cohen-Macalay varieties
zbMATH Open document ID: 0616.14039
DOI: 10.1007/BF01458428
author: Trung Ngo Viet
author: G. Valla
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164413
review text: \textit{R. Treger} [Duke Math. J. 48, 35-47 (1981; Zbl 0474.14030)] has shown that the defining prime ideal \(I_ V\) of an arithmetically Cohen- Macaulay non-degenerate variety V of degree s and codimension r may be generated by forms of degree \(\geq ]s/r[\), where \(]s/r[\) denotes the least integer \(\geq s/r\). This paper gives finer degree bounds for the generators of \(I_ V\) which involves the number of independent conditions imposed by V on the hypersurfaces of some fixed degree. As a consequence, we can describe almost all cases where Treger's bound is attained. Namely, suppose that \(s\geq (r+1)^ 2\), then \(I_ V\) needs a generator of degree \(]s/r[\) iff \(V\) lies on a variety of degree r and codimension \(r-1\) (variety of minimal degree), cut out by the quadric hypersurfaces containing V. In particular, if V is a smooth variety with \(s\geq \max \{dr+2,(r+1)^ 2\},\) \(d:=\dim(V)\), then \(I_ V\) needs a generator of degree ]s/r [ iff V is a Castelnuovo variety (i.e. of maximal geometric genus). If V does not lie on any hypersurface of degree t, we can show that \(I_ V\) may be generated by forms of degree \(\leq ]s/r[-[t(t-1)/4]\) (or \(\leq ]s/r[-t(r-1)/2\) if \(r\geq s\) or \(s\geq t^ 3- 3t-1)\).
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14M10
zbMATH DE Number: 3997994
zbMATH Keywords: degree bounds for the defining equations
zbMATH Keywords: defining prime ideal
zbMATH Keywords: arithmetically Cohen-Macaulay non-degenerate variety
zbMATH Keywords: Castelnuovo variety
MaRDI profile type: Publication
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OpenAlex ID: W2051968353
description: scientific article; zbMATH DE number 3997994