## An Invitation to Algebraic GeometryThe aim of this book is to describe the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. Few algebraic prerequisites are presumed beyond a basic course in linear algebra. |

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### Contents

Affine Algebraic Varieties | 1 |

11 Definition and Examples | 2 |

12 The Zariski Topology | 6 |

13 Morphisms of Affine Algebraic Varieties | 9 |

14 Dimension | 11 |

Algebraic Foundations | 15 |

22 Hilberts Basis Theorem | 18 |

23 Hilberts Nullstellensatz | 20 |

56 The Hilbert Function | 81 |

Smoothness | 85 |

62 Smooth Points | 92 |

63 Smoothness in Families | 96 |

64 Bertinis Theorem | 99 |

65 The Gauss Mapping | 102 |

Birational Geometry | 105 |

72 Rational Maps | 112 |

24 The Coordinate Ring | 23 |

25 The Equivalence of Algebra and Geometry | 26 |

26 The Spectrum of a Ring | 30 |

Projective Varieties | 33 |

32 Projective Varieties | 37 |

33 The Projective Closure of an Affine Variety | 41 |

34 Morphisms of Projective Varieties | 44 |

35 Automorphisms of Projective Space | 47 |

QuasiProjective Varieties | 51 |

42 A Basis for the Zariski Topology | 55 |

43 Regular Functions | 56 |

Classical Constructions | 63 |

52 Five Points Determine a Conic | 65 |

53 The Segre Map and Products of Varieties | 67 |

54 Grassmannians | 71 |

55 Degree | 74 |

73 Birational Equivalence | 114 |

74 Blowing Up Along an Ideal | 115 |

75 Hypersurfaces | 119 |

76 The Classification Problems | 120 |

Maps to Projective Space | 123 |

81 Embedding a Smooth Curve in Three Space | 124 |

82 Vector Bundles and Line Bundles | 127 |

83 The Sections of a Vector Bundle | 129 |

84 Examples of Vector Bundles | 131 |

85 Line Bundles and Rational Maps | 136 |

86 Very Ample Line Bundles | 141 |

Sheaves and Abstract Algebraic Varieties | 145 |

A2 Abstract Algebraic Varieties | 150 |

153 | |

157 | |

### Common terms and phrases

abstract algebraic variety affine algebraic variety affine space affine variety algebraic geometry automorphism birational equivalence blowup C-algebra C[Xi canonical bundle change of coordinates closed subset closed subvariety Cn+1 complex numbers cone conic coordinate ring corresponding defined definition degree denoted desingularization dimension divisors embedding Euclidean topology example Exercise fiber finitely Gauss map global sections Grassmannian Hilbert polynomial Hilbert scheme Hilbert's Nullstellensatz homogeneous coordinates homogeneous polynomials homomorphism hyperplane bundle hypersurface irreducible isomorphism line bundle linear subvariety matrix maximal ideal morphism morphism of algebraic nonzero open set open subset origin parabola plane polynomial function projective closure projective space projective variety projectively equivalent proof pullback quasi-projective variety radical ideal rational map regular functions restriction ringed spaces Segre map sheaf of sections sheaves Show subspace subvariety tangent bundle tangent space theorem theory topological space twisted cubic unique V C P vector bundle vector space Veronese map Zariski topology zero set

### References to this book

Computing in Algebraic Geometry: A Quick Start Using SINGULAR Wolfram Decker,Christoph Lossen Limited preview - 2006 |