Bounding cohomological Hilbert functions by hyperplane sections. (English) Zbl 0847.14007

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 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 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 in the proofs are new ideas from U. Nagel [J. Algebra 150, No. 1, 231-244 (1992; Zbl 0757.13004)]. The final section applies these bounds to vector bundles \({\mathcal E}\), obtaining bounds on the Chern classes of \({\mathcal E}\).


14F17 Vanishing theorems in algebraic geometry
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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