By Choudhary P.

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E. one generally considers only linear system having every member positive or null. If (A) is a point of A and J5 a point OfZ(A), not lying in eachFi(X) = 0, then ((A), J5) is a specialisation {see VAN DER WAERDEN [bJ, p. 182} of ((A), P) over k, P being a generic point of Z(A) over k(A): therefore the cycle Z in V X A associated with the cycle Z(A) by the relations Z(A) X X (A) = (V X (A))' Z and prvZ = V {see (I, 9)}, is a multiple 01 a k-variety and then so also is Z(A). Moreover, if P is a generic point of V over k, putting: P X A(P) = Z· (P X A), the cycle A(P) = rp-l(P) {for the straightforward definition of rp-l see (VI, 3)} must be a linear (m - 1)dimensional subvariety of A {see (I, 9) and use the fact that prLZ = L}: it follows that m generic points P v P 2 , • • • , Pm on V, independent over k, belong to one and only one cycle of L rational over k(P v ...

G. [6, 8, 10J}, where he used prevalently the quasigleichheit method, arrived at the quite simple treatment [2lJ: we recall also the work [IJ of FERNANDEZ BIARGE. We add that other proofs of BERTINI'S theorems have been given by VAN DER W AERDEN in [5, XJ and by B. SEGRE in [9]. Finally the c1assical notion of a linear system on a surface which is complete relatively to a set of points, either ordinary or infinitely near {see ZARISKI [aJ, p. 29}, has been extended to higher varieties by VAN DER WAERDEN in [10J by means of valuation theory, in such a manner as to satisfy the fundamental condition of being a birationally invariant notion.

Whenever a Q. T. creates some new multiple k-curve. Now let F, F v ... , F i , ... be a sequence of any birational transforms of Fand P, P v ... , Pi' ... a sequence of related corresponding k-points with Pi EF i. If Fi+l= Ti[Fi], the sequence {Pi} is called anormal sequence of'll-ple k-points if the following conditions are satisfied: 1. Every Pi is 'li-pIe for F i , and is either isolated on F i , or it is simple on a 'li-pIe k-curve or is anormal crossing of two 'li-pIe k-curves; 2. {Ti} is a normal sequence of permissible transformations.