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.


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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