Basic Algebraic Geometry, Volume 2
The second volume of Shafarevich's introductory book on algebraic varieties and complex manifolds. As with Volume 1, the author has revised the text and added new material, e.g. as a section on real algebraic curves. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum, making the book accessible to non-specialists. It can be read independently of Volume 1 and is suitable for beginning graduate students in mathematics as well as those in theoretical physics.
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2 Closed Subsets of Affine Spaces
3 Rational Functions
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affine affine variety algebraic algebraic curve algebraic variety apply arbitrary associated assume automorphism basis birational bundle called closed set complex components concept condition connected consider consists construct contained coordinates corresponding covering curve defined definition denote determines differential forms dimension divisor easy elements embedding equal equation equivalent example Exercise exists fact factor field finite follows functions geometry give given hence holds homomorphism ideal integral intersection invariant inverse image irreducible isomorphic Lemma linear means module morphism multiplication neighbourhood normal obtain Obviously open set parameters particular plane polynomial prime projective proof proposition proved quasiprojective rational functions reducible regular regular mapping relation restriction result ring satisfying scheme sheaf Show simple space Spec subset subvariety sufficient Suppose surface tangent Theorem theory topological unique vanish vector verify zero