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

Published online by Cambridge University Press:  22 January 2016

Claudia Albertini and
Markus Brodmann
Claudia Albertini
Affiliation:
Kantonsschule Zuercher Oberland, Fachkreis Mathematik, Fachkreis Mathematik 8620 Wetzikon, Switzerland
Markus Brodmann
Affiliation:
Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Züurich, Switzerland, brodmann@math.unizh.ch
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Abstract

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We consider a positively graded noetherian domain R = ⊕n∈NoRn 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>0Rm 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.

Keywords

13D0314B1514F1514F05

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 163 , September 2001, pp. 87 - 106
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

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