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Algebraic Geometry

By James D. Lewis

ISBN-10: 0821805681

ISBN-13: 9780821805688

This e-book presents an creation to a subject matter of principal curiosity in transcendental algebraic geometry: the Hodge conjecture. such as 15 lectures plus addenda and appendices, the amount relies on a sequence of lectures introduced via Professor Lewis on the Centre de Recherches Mathematiques (CRM). The e-book is a self-contained presentation, thoroughly dedicated to the Hodge conjecture and comparable themes. It contains many examples, and so much effects are thoroughly confirmed or sketched. the incentive in the back of the various effects and historical past fabric is supplied. This entire method of the ebook supplies it a ``user-friendly'' variety. Readers don't need to seek somewhere else for varied effects. The e-book is appropriate to be used as a textual content for a themes direction in algebraic geometry; comprises an appendix by means of B. Brent Gordon.

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Extra info for A survey of the Hodge conjecture

Sample text

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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