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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: (DLC) 2017045552 (OCoLC)1013506217 |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Steven Dale Cutkosky |

ISBN: | 9781470446703 1470446707 |

OCLC Number: | 1039613757 |

Description: | 1 online resource. |

Contents: | Cover; Title page; Contents; Preface; Chapter 1. A Crash Course in Commutative Algebra; 1.1. Basic algebra; 1.2. Field extensions; 1.3. Modules; 1.4. Localization; 1.5. Noetherian rings and factorization; 1.6. Primary decomposition; 1.7. Integral extensions; 1.8. Dimension; 1.9. Depth; 1.10. Normal rings and regular rings; Chapter 2. Affine Varieties; 2.1. Affine space and algebraic sets; 2.2. Regular functions and regular maps of affine algebraic sets; 2.3. Finite maps; 2.4. Dimension of algebraic sets; 2.5. Regular functions and regular maps of quasi-affine varieties. 2.6. Rational maps of affine varietiesChapter 3. Projective Varieties; 3.1. Standard graded algebras; 3.2. Projective varieties; 3.3. Grassmann varieties; 3.4. Regular functions and regular maps of quasi-projective varieties; Chapter 4. Regular and Rational Maps of Quasi-projective Varieties; 4.1. Criteria for regular maps; 4.2. Linear isomorphisms of projective space; 4.3. The Veronese embedding; 4.4. Rational maps of quasi-projective varieties; 4.5. Projection from a linear subspace; Chapter 5. Products; 5.1. Tensor products; 5.2. Products of varieties; 5.3. The Segre embedding. 5.4. Graphs of regular and rational mapsChapter 6. The Blow-up of an Ideal; 6.1. The blow-up of an ideal in an affine variety; 6.2. The blow-up of an ideal in a projective variety; Chapter 7. Finite Maps of Quasi-projective Varieties; 7.1. Affine and finite maps; 7.2. Finite maps; 7.3. Construction of the normalization; Chapter 8. Dimension of Quasi-projective Algebraic Sets; 8.1. Properties of dimension; 8.2. The theorem on dimension of fibers; Chapter 9. Zariski's Main Theorem; Chapter 10. Nonsingularity; 10.1. Regular parameters; 10.2. Local equations; 10.3. The tangent space. 10.4. Nonsingularity and the singular locus10.5. Applications to rational maps; 10.6. Factorization of birational regular maps of nonsingular surfaces; 10.7. Projective embedding of nonsingular varieties; 10.8. Complex manifolds; Chapter 11. Sheaves; 11.1. Limits; 11.2. Presheaves and sheaves; 11.3. Some sheaves associated to modules; 11.4. Quasi-coherent and coherent sheaves; 11.5. Constructions of sheaves from sheaves of modules; 11.6. Some theorems about coherent sheaves; Chapter 12. Applications to Regular and Rational Maps; 12.1. Blow-ups of ideal sheaves. 12.2. Resolution of singularities12.3. Valuations in algebraic geometry; 12.4. Factorization of birational maps; 12.5. Monomialization of maps; Chapter 13. Divisors; 13.1. Divisors and the class group; 13.2. The sheaf associated to a divisor; 13.3. Divisors associated to forms; 13.4. Calculation of some class groups; 13.5. The class group of a curve; 13.6. Divisors, rational maps, and linear systems; 13.7. Criteria for closed embeddings; 13.8. Invertible sheaves; 13.9. Transition functions; Chapter 14. Differential Forms and the Canonical Divisor; 14.1. Derivations and Kähler differentials. |

Series Title: | Graduate studies in mathematics, 188. |

Other Titles: | Algebraic geometry |

Responsibility: | Steven Dale Cutkosky. |

### Abstract:

Presents a readable and accessible introductory course in algebraic geometry, with most of the fundamental classical results presented with complete proofs. An emphasis is placed on developing connections between geometric and algebraic aspects of the theory. Differences between the theory in characteristic $0$ and positive characteristic are emphasized.
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