## 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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