The Geometry of Schemes
What schemes are The theory of schemes is the foundation for algebraic geometry for- lated by Alexandre Grothendieck and his many coworkers. It is the basis for a grand uni?cation of number theory and algebraic geometry, dreamt of by number theorists and geometers for over a century. It has streng- ened classical algebraic geometry by allowing ?exible geometric arguments about in?nitesimals and limits in a way that the classic theory could not handle. In both these ways it has made possible astonishing solutions of many concrete problems. On the number-theoretic side one may cite the proof of the Weil conjectures, Grothendieck’s original goal (Deligne ) and the proof of the Mordell Conjecture (Faltings ). In classical al- braic geometry one has the development of the theory of moduli of curves, including the resolution of the Brill–Noether–Petri problems, by Deligne, Mumford, Gri?ths, and their coworkers (see Harris and Morrison  for an account), leading to new insightseven in such basic areas as the t- ory of plane curves; the ?rm footing given to the classi?cation of algebraic surfaces in all characteristics (see Bombieri and Mumford ); and the development of higher-dimensional classi?cation theory by Mori and his coworkers (see Koll ́ ar ). No one can doubt the success and potency of the scheme-theoreticme- ods. Unfortunately, the average mathematician, and indeed many a - ginner in algebraic geometry, would consider our title, “The Geometry of Schemes”,anoxymoronakinto“civilwar”.
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