By Mario Baldassarri (auth.)
Algebraic geometry has continuously 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 through the Italian university which, by means of its pioneer paintings, in response to algebro-geometric equipment, has succeeded in build up an impressive physique of data. fairly except its intrinsic curiosity, this possesses excessive heuristic price because it represents a necessary step in the direction of the trendy achievements. a undeniable loss of rigour within the c1assical tools, in particular in regards to the principles, is essentially justified via the inventive impulse printed within the first phases of our topic; a similar phenomenon should be saw, to a better or much less volume, within the historic improvement of the other technological know-how, mathematical or non-mathematical. at the least, in the c1assical area itself, the principles have been later explored and consolidated, largely by way of SEVERI, on strains that have often 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, verified the equipment of contemporary summary algebraic geometry which, rejecting the c1assical restrict to the complicated groundfield, gave up geometrical instinct and undertook arithmetisation below the transforming into impact of summary algebra.
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Let X(it) be also the (r - I)-cycle on V defined by the divisor X(it): it is then obvious that as the it's vary in k the cycle X(it) varies in a linear system L associated with the module in k(V) (1, 11/10' ... , Im/lo). It is now easily proved, by the above representation, that the k-set of the points on V which lie in each IIX(it)ll, called the base set of L, is the set associated with the radical of the ideal (~o, ~1> ••• , ~m): moreover il a point Pis singular lor the sets IIXill, i = 0, 1, ...
The Geometrie Genus functionally independent over V'. Then the cycle determined by the system IW = 0, IW = 0 is defined: it contains the cycles D and 5; the residue, after these have been substracted, is still an (r - 1)dimensional cycle. This cycle is called the I aeobian eycle of Land is denoted by Cj if C is a member of L. b) Here we prove: (ii) The I aeobian eycles 01 alt the r-dimensional linear systems L eontained in a simple linear system L on V', belong to one and the same linear equivalenee class 01 V'.
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. The Canonical System as aBirational Invariant 41 depending only on T and not on the system L: thus the J acobian cycle r i is weIl defined. c) Suppose now that the system L is contained in a larger linear system L* satisfying the same hypotheses as in (b): by using the fact that, if L varies in L*, then the related cycle 'E does not vary, we obtain the following corollary to (i): (ii) The (complete) J acobian linear system 0/ the trans/ormed system on V 0/ a linear system L * on V, at least r-dimensional, simple and without base points, exists and is precisely that complete linear system containing the cycle r j = T[L i ] + E, where L is related to any r-dimensional subsystem L 0/ L*.