Algebraic Geometry and Arithmetic Curves (Oxford Graduate by Qing Liu,Reinie Erne PDF

By Qing Liu,Reinie Erne

ISBN-10: 0198502842

ISBN-13: 9780198502845

ISBN-10: 0199202494

ISBN-13: 9780199202492

This e-book is a basic creation to the idea of schemes, through purposes to mathematics surfaces and to the speculation of relief of algebraic curves.

The first half introduces simple items reminiscent of schemes, morphisms, base switch, neighborhood houses (normality, regularity, Zariski's major Theorem). this can be by means of 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 concept. the 1st half ends with the theory of Riemann-Roch and its software to the research of soft projective curves over a field.
Singular curves are handled via a close research 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 thought on mathematics surfaces. Castelnuovo's criterion is proved and in addition the lifestyles 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 e-book concludes with the funadmental theorem of good aid of Deligne-Mumford.

The e-book is basically self-contained, together with the mandatory fabric on commutative algebra. the must haves are hence few, and the e-book should still go well with a graduate pupil. It comprises many examples and approximately six hundred exercises.

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