Elementary Algebraic Geometry
American Mathematical Soc., 2003 - 213 sider
This is a genuine introduction to algebraic geometry. The author makes no assumption that readers know more than can be expected of a good undergraduate. He introduces fundamental concepts in a way that enables students to move on to a more advanced book or course that relies more heavily on commutative algebra. The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory.The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary. It is also an excellent text for those working in neighboring fields (algebraic topology, algebra, Lie groups, etc.) who need to know the basics of algebraic geometry.
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27 lines affine variety algebraic set algebraically closed assume base point free Bézout's theorem bijection birational called Chapter Co.g conic consider contained coordinate ring Corollary cubic curve cubic surface curve of degree defined Definition deg f dimension dom(f double point elements embedding equation Example field of fractions Figure genus geometric given gives hence homogeneous polynomials homomorphism hypersurface implies integral domain intersection inverse irreducible affine variety isomorphism k-algebra linear forms linear system map f matrix maximal ideal morphism MP/M multiplicity Noether normalization Noetherian nonempty nonzero open subset parametrization plane cubic plane curve polynomial map prime ideals principal divisor projective space projective variety proof of Proposition quadric quasi-affine variety quasi-projective variety radical ideal rational function rational map regular functions secant singular points smooth curve smooth point smooth projective curve subspace surjective tangent space transformation V X W Weierstraß Zariski topology zero