ON THE DEFINING FIELD OF A DIVISOR IN AN ALGEBRAIC VARIETY

In the system of algebraic geometry as developed by A. Weil in his recent book, Foundations of algebraic geometry, a variety U in the n-space is defined as the set of all equivalent couples (k, P), each consisting of a field k and a point P in the n-space such that the field k(P) is a regular extension of k. Two such couples (k', P') and (k″, P″) are called equivalent if every finite specialization of P' over k' is also one of P″ over k″ and conversely. Any field k which enters into such a couple is called a field of definition of the variety U. It has been shown by Weil in his book that among all the fields of definition of a variety U there is a smallest one which is contained in all of them, which we shall call the defining field of the variety U. A d-cycle G in a variety U of dimension r is a finite set of simple subvarieties of dimension d in U, to each of which is assigned an integer called its multiplicity; a cycle is called positive if the multiplicity of each of its component varieties is positive. Let K be a field of definition of U. Then the G is said to be rational over K if it satisfies the following conditions: (1) each component variety of G is algebraic over K; (2) if a variety is a component of G, then all the conjugate varieties over K are also components of G with the same multiplicity; (3) the multiplicity of each component of G is a multiple of its order of inseparability. The question arises whether there is a smallest one among all the fields over which the cycle G is rational. If such a smallest field exists, we shall call it the defining field of the cycle G . One observes that since by definition every field over which the cycle G is rational must be a field of definition of the variety U, it follows that the defining field of G, if it exists, must contain the defining field of U…