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

We consider a positively graded noetherian domain R = circle plus (n epsilo n N0) R-n for which Ro is essentially of finite type over a perfect field K of positive characteristic and we assume that the generic fibre of the nat ural morphism pi : Y = Proj(R) --> Y-0 = Spec(Ro) is geometrically connecte d, geometrically normal and of dimension > 1. Then we give bounds on the "r anks" of the n-th homogeneous part H-R+(2)(R)(n) of the second local cohomo logy module of R with respect to R+ := circle plus (m >0) R-m for n < 0. If Y is in addition normal, we shall see that the R-0-modules H-R+(2)(R)(n) a re torsion-free for all n < 0 and in this case our bounds on the ranks furn ish a vanishing result. From these results we get bounds on the first cohom ology of ample invertible sheaves in positive characteristic.