Basic Algebraic GeometrySpringer Science & Business Media, 6 déc. 2012 - 440 pages Algebraic geometry occupied a central place in the mathematics of the last century. The deepest results of Abel, Riemann, Weierstrass, many of the most important papers of Klein and Poincare belong to this do mam. At the end of the last and the beginning of the present century the attitude towards algebraic geometry changed abruptly. Around 1910 Klein wrote: "When I was a student, Abelian functions*-as an after-effect of Jacobi's tradition-were regarded as the undIsputed summit of mathe matics, and each of us, as a matter of course, had the ambition to forge ahead in this field. And now? The young generation hardly know what Abelian functions are." (Vorlesungen tiber die Entwicklung der Mathe matik im XIX. Jahrhundert, Springer-Verlag, Berlin 1926, Seite 312). The style of thinking that was fully developed in algebraic geometry at that time was too far removed from the set-theoretical and axio matic spirit, which then determined the development of mathematics. Several decades had to lapse before the rise of the theory of topolo gical, differentiable and complex manifolds, the general theory of fields, the theory of ideals in sufficiently general rings, and only then it became possible to construct algebraic geometry on the basis of the principles of set-theoretical mathematics. Around the middle of the present century algebraic geometry had undergone to a large extent such a reshaping process. As a result, it can again lay claim to the position it once occupied in mathematics |
Table des matières
3 | |
11 | |
Quasiprojective Varieties | 30 |
Regular Functions | 38 |
6 | 52 |
Chapter II | 71 |
Expansion in Power Series | 81 |
3 Properties of Simple Points | 90 |
Sheaves | 234 |
Products of Schemes | 254 |
Varieties | 263 |
2 Abstract and Quasiprojective Varieties | 282 |
3 Coherent Sheaves | 294 |
Topology of Algebraic Varieties | 308 |
3 The Topology of Algebraic Curves | 325 |
Real Algebraic Curves | 336 |
The Structure of Birational Isomorphisms | 98 |
Exceptional Subvarieties | 107 |
Divisors and Differential Forms | 127 |
Divisors on Curves | 140 |
Differential Forms | 156 |
Examples and Applications of Differential Forms | 166 |
Intersection Indices | 182 |
Applications and Generalizations of Intersection Indices | 198 |
Birational Isomorphisms of Surfaces | 208 |
Schemes | 221 |
Complex Analytic Manifolds | 343 |
2 Divisors and Meromorphic Functions | 360 |
3 Algebraic Varieties and Analytic Manifolds | 369 |
Uniformization | 380 |
409 | |
Geometry of Algebraic Curves | 421 |
Algebraic Varieties over an Arbitrary Field Schemes | 428 |
434 | |
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Expressions et termes fréquents
Abelian affine variety algebraic curve algebraic variety analytic manifold arbitrary automorphism birational isomorphism birationally isomorphic C₁ called closed sets closed subset codimension coherent sheaf compact complex numbers components concept condition consider consists construct contained coordinates Corollary corresponding D₁ defined definition degree denote determines differential forms dimension elements elliptic embedding equation equivalent example Exercise exists F₁ fact field functions f genus homomorphism integral intersection invariant inverse image irreducible variety Lemma linear local ring mapping f mapping ƒ matrix meromorphic module morphism multiplicity o-process obtain one-dimensional open set parameters plane point xe polynomial presheaf prime ideal projective space projective variety proof proposition proved quasiprojective variety rational functions rational mapping regular mapping scheme sheaf sheaves Show simple point singular points smooth projective curve smooth variety Spec subspace subvariety surface t₁ tangent space Theorem topological space triangulation u₁ vector bundle vector space X₁ y₁