By Qing Liu,Reinie Erne
The first half introduces simple gadgets resembling schemes, morphisms, base switch, neighborhood houses (normality, regularity, Zariski's major Theorem). this is often by means of the extra worldwide point: 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 learn of gentle projective curves over a field.
Singular curves are handled via an in depth examine of the Picard group.
The moment half starts off with blowing-ups and desingularisation (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 likewise the life of the minimum commonplace version. This ends up in the learn of aid of algebraic curves. The case of elliptic curves is studied intimately. The booklet concludes with the funadmental theorem of good aid of Deligne-Mumford.
The booklet is largely self-contained, together with the required fabric on commutative algebra. the must haves are accordingly few, and the e-book should still go well with a graduate scholar. It comprises many examples and approximately six hundred exercises.
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Additional info for Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics (0-19-961947-6))
Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics (0-19-961947-6)) by Qing Liu,Reinie Erne