The Nucleoprotein Content of Whole Nuclei

of A' over k.Since P and A' are centers of v, A' belongs to the set TIPI .Since P' is an isolated point of TIP), we must have dim A'7k = dim P'/k.From this it follows that A' and P' are isomorphic points over k (since P' is a specialization of A' over k) and that consequently the local rings of A' and P' coincide.Since o c o', the local field k(A') contains the local field k(P) of P, and hence A' is an algebraic point over k(P) (since dim A 'lk = dim P'/k = dim P/k).Since TIPi is a variety over k(P), we conclude that TIP) is zero dimensional over k(P) and therefore consists of a finite number of points.This completes the proof.Our "Main Theorem" on birational transformations follows from the above theorem if one takes into account that if V is locally normal at P then (a) V is analytically irreducible at P (Zariski4) and (b) the finite set TIP) necessarily consists of a single point (Zariski3, theorem 8(A) and theorem 10, pp.