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A priori bounds of Severi type for cohomological Hilbert-functions. (English) Zbl 0788.14012

Let \(F\) be a coherent sheaf on the projective \(d\)-space \(\mathbb{P}^ d\) and \(\delta(F):=\min\{\text{depth} F_ x=| x\in\mathbb{P}^ d\) closed point}. A well known result of Serre (which implies the lemma of Enriques-Severi-Zariski) says that, for \(i<\delta(F)\), \(H^ iF(n)=0\) for \(n\ll 0\). In this paper, the author gives a quantitative version of this vanishing theorem in terms of some invariants of the sheaf such as \(\delta(F)\), \(h^ 0F\), \(h^ 1F(-1),\dots,h^ iF(-i)\), the linear subdimension of \(F\) (a notion due to the author) defined as the minimal dimension of a linear subspace containing a point of \(\text{Ass}(F)\), etc. Since the bounds given by the author are expressed in a rather complicated way, we shall illustrate his results by stating a particular case: if \(0\leq i<\delta(F)\) and \(h^ jF(-j)=0\) for \(0\leq j\leq i\) then \(h^ iF(n)=0\) for all \(n\leq -i\). The main tool used by the author is the exact sequence of the hyperplane section.

MSC:

14F17 Vanishing theorems in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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