The hyperplane sections of normal varieties

Abstract

covered in that investigation show, however, that we have in them a class of varieties demanding study for its own sake. Another very good, but in one sense probably more transient, reason for the study of normal varieties is that as yet we are not assured of the existence of a model free from singularities for any given field of algebraic functions, and in fact a greater knowledge of normal varieties may be a prerequisite for the resolution of singularities of arbitrary varieties. Below we direct attention to the question whether, or to what extent, the hyperplane sections of a normal variety(2) are themselves normal. Quite generally, if P is a property of irreducible varieties, we may ask whether the hyperplane sections of a variety with property P share this property. In particular, we may raise this question for the property P of being irreducible. For curves, it is clear that the hyperplane sections will for the most part be reducible, so we shall confine the question to varieties of dimension r ?2. For varieties of dimension r> 2, it is still clear that not all the hyperplane sections will, in general, be irreducible: for example, consider a (suitable) cone; the hyperplane sections through the vertex will be reducible. This example leads us to reformulate the question. The hyperplanes of a projective space in themselves form a projective space, the dual space Sn': we shall say that almost all hyperplanes have the property P, if the hyperplanes not having the property P lie on (though they need not fill out) a proper algebraic subvariety of Sn'. Even if now it turned out to be false that almost all hyperplane sections of an irreducible variety are themselves irreducible, we would not consider the original question on normal varieties as closed, but would reformulate it in local terms; it tturns out, however, that they are irreducible almost always (Theorem 12), and therefore it is possible to deal with the