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Bounds on the cohomological Hilbert functions of a projective variety. (English) Zbl 0578.14015

We give bounds on the cohomological Hilbert functions \(h^ i_{{\mathcal F}}(n)=\dim_ kH^ i(X,{\mathcal F}(n))\), where \({\mathcal F}\) is a coherent sheaf over a projective variety over an algebraically closed field k. These bounds are given in terms of the functions \(h^ i_{{\mathcal F}| H}\), where H runs through a linear system of hyperplane sections of X which is in general position.
Applying these bounds to vector bundles over projective spaces, we get estimates on their cohomological Hilbert functions which depend only from the first two Chern classes and the ”span” of the generic splitting type. Similar bounds have been given by G. Elencwajg and O. Forster [Math. Ann. 246, 251-270 (1980; Zbl 0432.14011)]. - Applying our general method to a very ample divisor \({\mathcal L}\) over a complete irreducible variety X, we obtain a bound \(n_ 0\) such that \(h^ 1(X,{\mathcal L}^ n)\) takes the same value for all \(n\leq n_ 0\). Thereby \(n_ 0\) depends only on \(h^ 1(X,{\mathcal O}_ X)\) and on local data of X, but not on the characteristic of the ground field k.

MSC:

14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14C05 Parametrization (Chow and Hilbert schemes)

Citations:

Zbl 0432.14011
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References:

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