Abstract
We consider a positively graded noetherian domain $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ for which $R_{0}$ is essentially of finite type over a perfect field $K$ of positive characteristic and we assume that the generic fibre of the natural morphism $\pi : Y = \rm{Proj}(R) \to Y_{0} = \rm{Spec}(R_{0})$ is geometrically connected, geometrically normal and of dimension $> 1$. Then we give bounds on the "ranks" of the $n$-th homogeneous part $H^{2}_{R_{+}} (R)_{n}$ of the second local cohomology module of $R$ with respect to $R_{+} := \bigoplus_{m > 0} R_{m}$ 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.
Citation
Claudia Albertini. Markus Brodmann. "A bound on certain local cohomology modules and application to ample divisors." Nagoya Math. J. 163 87 - 106, 2001.
Information