By Qing Liu

This new-in-paperback variation offers a normal advent to algebraic and mathematics geometry, beginning with the idea of schemes, through functions to mathematics surfaces and to the idea of aid of algebraic curves.

The first half introduces simple items reminiscent of schemes, morphisms, base switch, neighborhood homes (normality, regularity, Zariski's major Theorem). this can be via the extra international 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 conception. the 1st half ends with the concept of Riemann-Roch and its software to the research of gentle projective curves over a box. Singular curves are taken care of via an in depth research 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 conception on mathematics surfaces. Castelnuovo's criterion is proved and in addition the lifestyles of the minimum average version. This results in the examine of relief of algebraic curves. The case of elliptic curves is studied intimately. The e-book concludes with the elemental theorem of reliable relief of Deligne-Mumford.

This booklet is largely self-contained, together with the required fabric on commutative algebra. the must haves are few, and together with many examples and nearly six hundred workouts, the e-book is perfect for graduate students.

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