By Mario Baldassarri (auth.)

Algebraic geometry has continually been an ec1ectic technological know-how, with its roots in algebra, function-theory and topology. except early resear ches, now a few century previous, this pretty department of arithmetic has for a few years been investigated mainly by means of the Italian institution which, through its pioneer paintings, according to algebro-geometric tools, has succeeded in increase an impressive physique of data. rather except its intrinsic curiosity, this possesses excessive heuristic price because it represents a vital step in the direction of the fashionable achievements. a definite loss of rigour within the c1assical tools, specifically in regards to the rules, is essentially justified through the inventive impulse printed within the first phases of our topic; a similar phenomenon may be saw, to a better or much less volume, within the old improvement of the other technology, mathematical or non-mathematical. at least, in the c1assical area itself, the principles have been later explored and consolidated, largely through 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 faculties, confirmed the equipment of contemporary summary algebraic geometry which, rejecting the c1assical limit to the complicated groundfield, gave up geometrical instinct and undertook arithmetisation below the growing to be effect of summary algebra.

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One can now show, by a simple calculation {see SEVERI [35J, pp. 10 to 13}, that J(~) = (1 + Nl) g(~)T+l. IW on V'. But the form g(~) = 0 has with V', along E and D ' , intersection multiplicity unity, and, therefore, 1W = 0 cuts on V' a cyc1e which, residually to the components 5, D and D ', is the sum of the two cyc1es (r + 1) E and Cj' We observe that the cycle E might possibly present itself more than (r + 1) times in (C + E)j: this will obviously happen if and only if E is already a component of C j' d) The following is the ENRIQUES adjunetion theorem: (iv) Let ICll and IC 2 1 be two simple linear systems, at least r-dimensional, on V'.

V r) respectively as uniformisingparameters on Vand Vat P and P. The correspondence T can be locally represented by the equations: Vi = fi(U) and the inverse T-l by Ui = gi(V), (i = 1, ... , r). The functions fi and gi are meromorphic functions of the arguments in some neighbourhood of P or of P respectively: more precisely they are holomorphic, provided T be biregular at the pair (P, P). This is certainly the case if Pis not fundamental, Vand V being here non-singular. 40 IV. The Geometrie Genus Suppose now that P be fundamental for T and P non-fundamental for T-l: then the determinant J(v) = o(g)jo(v) is itself holomorphic locally at P, vanishing at P, because otherwise T would be biregular at (P, P).

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. IV. The Geometrie Genus 1. The Adjoint Forms We now give an aeeount of the classical theory of the geometrie genus for a variety defined over the complex lield, following a reeent exposition by SEVERI {see SEVERI [35]}: later on we shall give a treatment of the same subject, over any field k, from a different standpoint {see (IV,4)}.