A bound on certain local cohomology modules and application to ample divisors

We consider a positively graded noetherian domain R = ⊕n∈No Rn for which R0 is essentially of finite type over a perfect field K of positive characteristic and we assume that the generic fibre of the natural morphism π: Y = Proj(R) → Y 0 = Spec(R 0) is geometrically connected, geometrically normal a...

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Bibliographic Details
Published in:Nagoya mathematical journal 2001-09, Vol.163, p.87-106
Main Authors: Albertini, Claudia, Brodmann, Markus
Format: Article
Language:English
Subjects:
13D45
14B15
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Summary:We consider a positively graded noetherian domain R = ⊕n∈No Rn for which R0 is essentially of finite type over a perfect field K of positive characteristic and we assume that the generic fibre of the natural morphism π: Y = Proj(R) → Y 0 = Spec(R 0) is geometrically connected, geometrically normal and of dimension > 1. Then we give bounds on the “ranks” of the n-th homogeneous part H 2(R)n of the second local cohomology module of R with respect to R +:= ⊕m>0 Rm for n < 0. If Y is in addition normal, we shall see that the R 0-modules H 2 R + (R)n are torsion-free for all n < 0 and in this case our bounds on the ranks furnish a vanishing result. From these results we get bounds on the first cohomology of ample invertible sheaves in positive characteristic.
ISSN:0027-7630
2152-6842
DOI:10.1017/S0027763000007923