Canonical systems on a reducible variety

Abstract

The object of this note is to extend the proof, for the general canonical varieties of any dimension k < d on a Vd whose existence was established by Dr Todd in a recent paper, of the property of adjunction given by the relationwhere S is any non-singular variety of dimension d−1 lying on Vd, and Xk[S] denotes the canonical system of dimension k on S. This formula was proved by Todd in the case when Vd is a general non-singular variety, and S is non-singular and belongs to a sufficiently general linear system. Using this result, I show how to remove the latter restriction. (1·1) cannot be taken, without further proof, as defining canonical systems on virtual or isolated varieties, as is done by B. Segre, since the canonical systems on an effective isolated variety are already well defined.