By Mario Baldassarri (auth.)

Algebraic geometry has regularly been an ec1ectic technology, with its roots in algebra, function-theory and topology. except early resear ches, now a couple of century previous, this gorgeous department of arithmetic has for a few years been investigated mainly via the Italian institution which, by way of its pioneer paintings, in response to algebro-geometric equipment, has succeeded in build up a majestic physique of data. relatively except its intrinsic curiosity, this possesses excessive heuristic worth because it represents an important step in the direction of the fashionable achievements. a undeniable loss of rigour within the c1assical tools, particularly in regards to the principles, is basically justified by way of the artistic impulse published within the first levels of our topic; an analogous phenomenon may be saw, to a better or much less quantity, within the historic improvement of the other technology, mathematical or non-mathematical. as a minimum, in the c1assical area itself, the rules have been later explored and consolidated, largely via SEVERI, on traces that have usually encouraged extra investigations within the summary box. approximately twenty-five years in the past B. L. VAN DER WAERDEN and, later, O. ZARISKI and A. WEIL, including their faculties, proven the equipment of contemporary summary algebraic geometry which, rejecting the c1assical restrict to the advanced groundfield, gave up geometrical instinct and undertook arithmetisation below the transforming into impact of summary algebra.

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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.