By Qing Liu,Reinie Erne
The first half introduces easy gadgets comparable to schemes, morphisms, base switch, neighborhood homes (normality, regularity, Zariski's major Theorem). this is often by means of the extra worldwide 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 idea. the 1st half ends with the concept of Riemann-Roch and its software to the learn of tender projective curves over a field.
Singular curves are handled via a close learn of the Picard group.
The moment half begins with blowing-ups and desingularisation (embedded or no longer) 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 general version. This ends up in the learn of aid of algebraic curves. The case of elliptic curves is studied intimately. The e-book concludes with the funadmental theorem of sturdy aid of Deligne-Mumford.
The ebook is basically self-contained, together with the mandatory fabric on commutative algebra. the necessities are hence few, and the ebook should still swimsuit a graduate scholar. It includes 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