Bounding cohomological Hilbert functions by hyperplane sections
type: Item
instance of: scholarly article
title: Bounding cohomological Hilbert functions by hyperplane sections
zbMATH Open document ID: 0847.14007
author: Uwe Nagel
author: M. P. Brodmann
published in: Journal of Algebra
publication date: 1995-08-29T00:00:00Z
review text: Let \(X \subseteq \mathbb{P}^r_k\) be a projective variety, \({\mathcal F}\) be a coherent sheaf of \({\mathcal O}_X\)-modules, and \(h^i_{\mathcal F} (t) = \dim_k H^i (X, {\mathcal F} (t))\) \((i \geq 0\), \(k\) the ground field). If \(i > 0\) then there exists \(N\) depending on \({\mathcal F}\) such that \(h^i_{\mathcal F} (t) = 0\) for all \(t \geq N\) [a theorem of Castelnuovo-Serre; see for example section 66 of \textit{J.-P. Serre}, ``Faisceaux algébriques cohérents'', Ann. Math., II. Ser. 61, 197-278 (1955; Zbl 0067.16201)] and whenever \(i < \delta ({\mathcal F})\) then there exists an integer \(M\) depending on \({\mathcal F}\) such that \(h^i_{\mathcal F} (t) = 0\) for all \(t \leq - M\) [a theorem of Severi, Enriques, Zariski, and Serre; see for example \textit{J.-P. Serre} (loc. cit.), section 74]. Here \(\delta ({\mathcal F})\) is the global subdepth of \({\mathcal F}\). The goal of the authors is to give bounds (which they call ``a priori bounds'') on \(N\) and \(M\) which depend on a finite number of invariants of the pair \((X, {\mathcal F})\). The bounds obtained in the present paper improve on those obtained in a series of papers by the first author, beginning with Comment. Math. Helv. 65, No. 3, 478-518 (1990; Zbl 0728.14014). The invariants used include the dimension \(\dim ({\mathcal F}) (= \dim \text{supp} ({\mathcal F}))\) of \({\mathcal F}\), \(\delta ({\mathcal F})\), the linear subdimension \(\text{lsdim} ({\mathcal F})\) of \({\mathcal F}\) and certain values of \(h^j_{\mathcal F} (-j)\). (These invariants are explained in the present paper, and in earlier papers by the first author.) The bounds are obtained in the form of recursive formulas, which are used to compute several examples. An important ingredient of the proofs is the method of ``hyperplane sections'' which uses the behavior of \(h^i_{{\mathcal F} |H}\) \((H\) ranging over sufficiently general hyperplanes) to say something about the behavior of \(h^i_{\mathcal F}\). Also used …
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 14C17
zbMATH DE Number: 783867
zbMATH Keywords: bounds for cohomological Hilbert functions
zbMATH Keywords: hyperplane sections
zbMATH Keywords: Chern classes
reviewed by: Leslie G. Roberts
MaRDI profile type: Publication
full work available at URL: jabr.1995.1130
OpenAlex ID: W1993821664
DOI: 10.1006/JABR.1995.1130
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description: scientific article; zbMATH DE number 783867