By Qing Liu

This new-in-paperback variation presents a common creation to algebraic and mathematics geometry, beginning with the idea of schemes, through purposes to mathematics surfaces and to the idea of relief of algebraic curves.

The first half introduces uncomplicated gadgets resembling schemes, morphisms, base swap, neighborhood houses (normality, regularity, Zariski's major Theorem). this is often through 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 thought. the 1st half ends with the concept of Riemann-Roch and its program to the learn of gentle projective curves over a box. Singular curves are handled via a close research of the Picard group.

The moment half begins 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 likewise the life of the minimum usual version. This ends up in the research of relief of algebraic curves. The case of elliptic curves is studied intimately. The booklet concludes with the basic theorem of strong relief of Deligne-Mumford.

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

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