A well known theorem of Serre
states the equivalence between the ampleness of a linear equivalence class of divisors
on an algebraic variety and the vanishing of the first cohomology groups related to
sufficiently high multiples of such linear equivalence class. In this paper the result of
the above theorem is extended in the following direction: given a linear equivalence
class on an algebraic variety, does there exist a cohomological characterization of the
open subset consisting of points of the variety which belong to affine open
complements of effective divisors in the multiples of the given class? The
characterization obtained is the main result, and it gives easily Serre’s result as a
particular case. While in one direction the proof uses the vanishing theorem quoted
in the beginning, it is independent of it in the opposite direction. A simple
application of the main result gives a first cohomological characterization of divisorial
varieties.
