Mumford D. - Curves and their jacobians

Curves and their jacobians
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This book is a slightly expanded version of the series of four Ziwet Lectures which I gave in November 1974 at The University of Michigan. Ann Arbor. The aim of the lectures and of this volume is to introduce people in the mathematical community at large-professors in other fields and graduate students beyond the basic courses-to what I find one of the most beautiful and what objectively speaking is at least one of the oldest topics in algebraic geometry: curves and their Jacobians. Because of time constraints, 1 had to avoid digressions on any засадничі topics and to rely on the standard definitions and intuitions of mathematicians in general. This is not always simple in algebraic geometry since its засадничі systems have tended to be more abstract and apparently more idiosyncratic than in other fields such as differential or analytic goemetry, and therefore not have become widely known to non-specialists. My idea was to get around this problem by imitating history: i.e., by introducing all the characters simultaneously in their complex analytic and algebraic forms. This did mean that I had to omit discussion of the characteristic p and arithmetic sides. However it also meant that 1 could immediately compare the strictly analytic constructions (such as Teichmuller Space) with the varieties which we were principally discussing. When I first started doing research in algebraic geometry, I thought the subject attractive for two reasons: firstly, because it dealt with such down-to-earth and really concrete objects as projective curves and surfaces; по-друге, because it was a small, quiet field where a dozen people did not leap on each new idea the minute it became current. As it turned out. the field seems to have acquired the reputation of being esoteric, exclusive and very abstract with adherents who are secretly plotting to take over all the rest of mathematics! In one respect this last point is accurate: algebraic geometry is a subject which relates frequently with a very large number of other fields-analytic and differential geometry, topology, k-theory, commutative algebra, algebraic groups and number theory, for instance-and both and gives receives theorems, techniques and examples with all of them. And certainly Grothendieck's work contributed to the field some very abstract and very powerful ideas which are quite hard to digest. But this subject, like all subjects, has a dual aspect in that all these abstract ideas would callapse of their own weight it were not for the underpinning supplied by concrete classical geometry. For me it has been a real ad\enture to perceive the interactions aspects of all these and to learn as much as 1 could about the theorems both old and new of algebraic geometry..