An Application of a Log Version of the Kodaira Vanishing Theorem to Embedded Projective Varieties

The idea in this paper is to show how a generalized “log” version of the Kodaira vanishing theorem can be employed to improve the results of Theorem 1 (which was also proved by Kodaira vanishing) when we have more knowledge about the equations for Y . The idea is to find a hypersurface F ⊂ P which has high multiplicity along Y , is “log canonical” near Y , and has relatively small degree, then to invoke Kodaira vanishing on the blow-up of P along Y . In the context of Theorem 1, the hypersurface F is approximately a divisor with normal crosssings (see its proof in §2). However one of the main points of this paper is the observation that even in the most familiar of projective embeddings, log canonical divisors quite different from normal crossings divisors seem to play an important role.