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Introduction to Algebraic Geometry
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Brendan Hassett
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Cambridge
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1 edition
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252
ISBN
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0521691419
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Algebraic geometry, central to pure mathematics, has important applications in such fields as engineering, computer science, statistics and computational biology, which exploit the computational algorithms that the theory provides. Users get the full benefit, however, when they know something of the underlying theory, as well as basic procedures and facts. This book is a systematic introduction to the central concepts of algebraic geometry most useful for computation. Written for advanced undergraduate and graduate students in mathematics and researchers in application areas, it focuses on specific examples and restricts development of formalism to what is needed to address these examples. In particular, it introduces the notion of Gr?ner bases early on and develops algorithms for almost everything covered. It is based on courses given over the past five years in a large interdisciplinary programme in computational algebraic geometry at Rice University, spanning mathematics, computer science, biomathematics and bioinformatics.
Preface xi
Guiding problems 1(10)
Implicitization 1(3)
Ideal membership 4(1)
Interpolation 5(3)
Exercises 8(3)
Division algorithm and Grobner bases 11(22)
Monomial orders 11(2)
Grobner bases and the division algorithm 13(3)
Normal forms 16(3)
Existence and chain conditions 19(3)
Buchberger\'s Criterion 22(4)
Syzygies 26(3)
Exercises 29(4)
Affine varieties 33(24)
Ideals and varieties 33(5)
Closed sets and the Zariski topology 38(1)
Coordinate rings and morphisms 39(4)
Rational maps 43(3)
Resolving rational maps 46(4)
Rational and unirational varieties 50(3)
Exercises 53(4)
Elimination 57(16)
Projections and graphs 57(4)
Images of rational maps 61(4)
Secant varieties, joins, and scrolls 65(3)
Exercises 68(5)
Resultants 73(16)
Common roots of univariate polynomials 73(7)
The resultant as a function of the roots 80(2)
Resultants and elimination theory 82(2)
Remarks on higher-dimensional resultants 84(3)
Exercises 87(2)
Irreducible varieties 89(12)
Existence of the decomposition 90(1)
Irreducibility and domains 91(1)
Dominant morphisms 92(2)
Algorithms for intersections of ideals 94(2)
Domains and field extensions 96(2)
Exercises 98(3)
Nullstellensatz 101(15)
Statement of the Nullstellensatz 102(1)
Classification of maximal ideals 103(1)
Transcendence bases 104(2)
Integral elements 106(2)
Proof of Nullstellensatz I 108(1)
Applications 109(2)
Dimension 111(1)
Exercises 112(4)
Primary decomposition 116(18)
Irreducible ideals 116(2)
Quotient ideals 118(1)
Primary ideals 119(3)
Uniqueness of primary decomposition 122(6)
An application to rational maps 128(3)
Exercises 131(3)
Projective geometry 134(35)
Introduction to projective space 134(3)
Homogenization and dehomogenization 137(3)
Projective varieties 140(1)
Equations for projective varieties 141(3)
Projective Nullstellensatz 144(1)
Morphisms of projective varieties 145(9)
Products 154(2)
Abstract varieties 156(6)
Exercises 162(7)
Projective elimination theory 169(12)
Homogeneous equations revisited 170(1)
Projective elimination ideals 171(3)
Computing the projective elimination ideal 174(1)
Images of projective varieties are closed 175(1)
Further elimination results 176(1)
Exercises 177(4)
Parametrizing linear subspaces 181(26)
Dual projective spaces 181(1)
Tangent spaces and dual varieties 182(5)
Grassmannians: Abstract approach 187(4)
Exterior algebra 191(6)
Grassmannians as projective varieties 197(2)
Equations for the Grassmannian 199(3)
Exercises 202(5)
Hilbert polynomials and the Bezout Theorem 207(28)
Hilbert functions defined 207(4)
Hilbert polynomials and algorithms 211(4)
Intersection multiplicities 215(4)
Bezout Theorem 219(6)
Interpolation problems revisited 225(4)
Classification of projective varieties 229(2)
Exercises 231(4)
Appendix A Notions from abstract algebra 235(11)
Rings and homomorphisms 235(1)
Constructing new rings from old 236(2)
Modules 238(1)
Prime and maximal ideals 239(1)
Factorization of polynomials 240(2)
Field extensions 242(2)
Exercises 244(2)
Bibliography 246(3)
Index 249
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