Vol. 37, No. 3, 1971

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The cohomology of divisorial varieties

Mario Borelli

Vol. 37 (1971), No. 3, 617–623

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.

Mathematical Subject Classification 2000
Primary: 14C20
Secondary: 14C10
Received: 23 September 1969
Revised: 25 March 1970
Published: 1 June 1971
Mario Borelli