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Algebraic Varieties by Mario Baldassarri (auth.)

By Mario Baldassarri (auth.)

Algebraic geometry has regularly been an ec1ectic technological know-how, with its roots in algebra, function-theory and topology. except early resear­ ches, now a few century outdated, this gorgeous department of arithmetic has for a few years been investigated mainly by way of the Italian university which, by means of its pioneer paintings, according to algebro-geometric tools, has succeeded in build up an impressive physique of data. relatively except its intrinsic curiosity, this possesses excessive heuristic price because it represents a vital step in the direction of the fashionable achievements. a undeniable loss of rigour within the c1assical equipment, specially with reference to the principles, is basically justified by means of the artistic impulse published within the first phases of our topic; an analogous phenomenon will be saw, to a better or much less volume, within the old improvement of the other technological know-how, mathematical or non-mathematical. at least, in the c1assical area itself, the rules have been later explored and consolidated, largely via SEVERI, on traces that have often encouraged additional 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 colleges, proven the tools of contemporary summary algebraic geometry which, rejecting the c1assical limit to the complicated groundfield, gave up geometrical instinct and undertook arithmetisation less than the starting to be impression of summary algebra.

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E Ai CfJi(g(V)) = 0, i=O and the determinant ](v), analogous to J(u) , associated with this system will be meromorphic locally at P and related to J(u) by the usual transformation law: 7(v) = J(u) . o(v)jo(u), which holds for each biregular pair of points for F, in the neighbourhoods of P and P. Recalling the result in a), we deduce at once from this relation that each analytic branch of an exceptional hypersurface of V through P, is contained in the analytical set 7(v) = 0, i. e. in some component of the J acobian cycle L; of L: in conclusion, this cycle contains all the exceptional (r - l)-dimensional hypersurfaces of 17 for T, and with a multiplicity completely determined by the order of o(v)jo(u) at P and therefore 3.

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 ...

A) Webegin with the theorem: (i) Let L be a simple r-dimensionallinear system on the hypersurlace V': then the locus 01 singular points belonging to the cycles 01 L is the support 01 an (r - 1)-dimensional cycle. For the proof let F(~) = 0 be the equation of V' in pr+l, and . E Ai FM) = 0 i=O be the equation of a linear system of adjoint forms r:p of order m, and such that: r:p. V' - 5 - D = G, where Gis any member of Land Da convenient fixed cycle on V'. The existence of such a system of adjoints follows at once from the completeness theorem and from the residue theorem {see (111,4) and (IV, I)}.

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