Algebraic GeometryRobin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. After receiving his Ph.D. from Princeton in 1963, Hartshorne became a Junior Fellow at Harvard, then taught there for several years. In 1972 he moved to California where he is now Professor at the University of California at Berkeley. He is the author of "Residues and Duality" (1966), "Foundations of Projective Geometry (1968), "Ample Subvarieties of Algebraic Varieties" (1970), and numerous research titles. His current research interest is the geometry of projective varieties and vector bundles. He has been a visiting professor at the College de France and at Kyoto University, where he gave lectures in French and in Japanese, respectively. Professor Hartshorne is married to Edie Churchill, educator and psychotherapist, and has two sons. He has travelled widely, speaks several foreign languages, and is an experienced mountain climber. He is also an accomplished amateur musician: he has played the flute for many years, and during his last visit to Kyoto he began studying the shakuhachi. |
Contents
14 | |
Nonsingular Varieties | 31 |
Intersections in Projective Space | 47 |
CHAPTER II | 60 |
First Properties of Schemes | 82 |
Separated and Proper Morphisms | 95 |
Sheaves of Modules | 108 |
Divisors | 129 |
The Canonical Embedding | 340 |
Classification of Curves in P | 349 |
CHAPTER V | 356 |
Geometry on a Surface | 357 |
Ruled Surfaces | 369 |
Monoidal Transformations | 386 |
The Cubic Surface in P3 | 395 |
Birational Transformations | 409 |
Projective Morphisms | 149 |
Differentials | 172 |
Formal Schemes | 190 |
Cohomology of Sheaves | 206 |
Cech Cohomology | 219 |
The Cohomology of Projective Space | 225 |
Ext Groups and Sheaves | 233 |
The Serre Duality Theorem | 239 |
Higher Direct Images of Sheaves | 250 |
Flat Morphisms | 253 |
Smooth Morphisms | 268 |
The Theorem on Formal Functions | 276 |
The Semicontinuity Theorem | 281 |
CHAPTER IV | 293 |
RiemannRoch Theorem | 294 |
Hurwitzs Theorem | 299 |
Embeddings in Projective Space | 307 |
Elliptic Curves | 316 |
Classification of Surfaces | 421 |
APPENDIX A Intersection Theory | 424 |
Intersection Theory | 425 |
Properties of the Chow Ring | 428 |
Chern Classes | 429 |
The RiemannRoch Theorem | 431 |
Complements and Generalizations | 434 |
APPENDIX B Transcendental Methods | 438 |
Comparison of the Algebraic and Analytic Categories | 440 |
When is a Compact Complex Manifold Algebraic? | 441 |
Kähler Manifolds | 445 |
The Exponential Sequence | 446 |
APPENDIX C | 449 |
Bibliography | 459 |
Results from Algebra | 470 |
478 | |
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Common terms and phrases
abelian groups algebraically closed field ample automorphism base points birational Cartier divisor closed immersion closed point closed subscheme closed subset coherent sheaf cohomology corresponding curve of degree curve of genus define definition denote dimension elements elliptic curve embedding equivalent exact sequence Example fibre finite morphism finite type flasque flat follows function field functor genus g gives global sections Hence homogeneous homomorphism induced injective integral invertible sheaf isomorphism Lemma let F Let f:X linear system locally free sheaf maximal ideal module multiplicity natural map noetherian ring noetherian scheme nonsingular curve nonsingular projective open affine subset open set open subset Ox-modules polynomial prime ideal Proj projective space projective variety PROOF Proposition quadric quasi-coherent quasi-coherent sheaf quotient rational Riemann-Roch ringed space ruled surface sheaf F sheaves singular Spec surjective tangent theorem topological space unique valuation ring X₁