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Algebraic Geometry

By I.G. Macdonald

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Taking A = X, we obtain CPx == 4~, which is the functional counterpart of PANNELLI'S second relation, W = 4 Q o + 1. Thus, in the present theory, PANNELLI'S equations appear as numerical aspects of equivalence relations which are themselves merely special cases of general functional results; we shall see shortly that SEVERI'S equation is capable of a similar interpretation. Writing the above equivalence in the form CPA == (A3) + 2 (A2X) + (AX2), we observe that it enables us to define the canonical series ICPAI in all cases, even when A is reducible or virtual.

More precisely, in the case d = 3, if there is a congruence of curves, its irregularity is at least q - Pg (;;; 3), while if there is a pencil of surfaces, its genus is at least q - Pg - 1 (;;; 2). COMESSATTI'S demonstration, like CASTELNUOVO'S, is transcendental, but, as D'ORGEVAL [1J has observed in the case d = 2, a geometrical proof may be given. Following CASTELNUOVO, we consider on ~ a continuous system {C} of hypersurfaces consisting of ooq disequivalent linear systems: such systems can always be found on Va.

Moreover, it depends only on

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