By Qing Liu
This new-in-paperback version offers a normal advent to algebraic and mathematics geometry, beginning with the idea of schemes, by means of purposes to mathematics surfaces and to the idea of aid of algebraic curves.
The first half introduces simple gadgets resembling schemes, morphisms, base swap, neighborhood houses (normality, regularity, Zariski's major Theorem). this can be through the extra international element: coherent sheaves and a finiteness theorem for his or her cohomology teams. Then follows a bankruptcy on sheaves of differentials, dualizing sheaves, and Grothendieck's duality thought. the 1st half ends with the theory of Riemann-Roch and its program to the examine of gentle projective curves over a box. Singular curves are handled via an in depth examine of the Picard group.
The moment half starts off with blowing-ups and desingularization (embedded or now not) of fibered surfaces over a Dedekind ring that leads directly to intersection idea on mathematics surfaces. Castelnuovo's criterion is proved and in addition the life of the minimum common version. This ends up in the research of relief of algebraic curves. The case of elliptic curves is studied intimately. The e-book concludes with the elemental theorem of strong aid of Deligne-Mumford.
This publication is largely self-contained, together with the required fabric on commutative algebra. the necessities are few, and together with many examples and nearly six hundred routines, the ebook is perfect for graduate students.
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