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First Course In Algebraic Geometry And Algebraic Varieties, A [Minkštas viršelis]

(Univ Of Rome Tor Vergata, Italy)
  • Formatas: Paperback / softback, 328 pages
  • Serija: Essential Textbooks in Mathematics
  • Išleidimo metai: 07-Mar-2023
  • Leidėjas: World Scientific Europe Ltd
  • ISBN-10: 1800612745
  • ISBN-13: 9781800612747
Kitos knygos pagal šią temą:
First Course In Algebraic Geometry And Algebraic Varieties, ADidesnis paveikslėlis
  • Formatas: Paperback / softback, 328 pages
  • Serija: Essential Textbooks in Mathematics
  • Išleidimo metai: 07-Mar-2023
  • Leidėjas: World Scientific Europe Ltd
  • ISBN-10: 1800612745
  • ISBN-13: 9781800612747
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781800612747.html
Raktažodžiai:

This Book Provides A Gentle Introduction To The Foundations Of Algebraic Geometry, Starting From Computational Topics (Ideals And Homogeneous Ideals, Zero Loci Of Ideals) Up To Increasingly Intrinsic And Abstract Arguments, Like ""Algebraic Varieties"", Whose Natural Continuation Is A More Advanced Course On The Theory Of Schemes, Vector Bundles And Sheaf-Cohomology. Valuable To The Students Studying In Algebraic Geometry And Geometry, A First Course In Algebraic Geometry And Algebraic Varieties Contains About 60 Solved Exercises To Help Students Thoroughly Understand The Theories Introduced In The Book. Proofs Of The Results Are Carried Out In Full Details. Many Examples Are Discussed Which Reinforces The Understanding Of Both The Theoretical Elements, The Consequences And The Possible Applications Of The Material.

Preface vii
About the Author xi
Acknowledgments xiii
1 Basics on Commutative Algebra
1(54)
1.1 Ideals and Operations on Ideals
2(2)
1.2 UFDs and PIDs
4(1)
1.3 Polynomial Rings
5(8)
1.3.1 Polynomials in D[ x], where D a UFD
5(3)
1.3.2 The case D = Ka field
8(1)
1.3.3 Resultant of two polynomials in D[ x]
9(3)
1.3.4 Resultant in D[ x1,...,xn] and elimination
12(1)
1.4 Noetherian Rings and the Hilbert Basis Theorem
13(3)
1.5 R-Modules, R-Algebras and Finiteness Conditions
16(3)
1.6 Integrality
19(2)
1.7 Zariski's Lemma
21(2)
1.8 Transcendence Degree
23(4)
1.9 Tensor Products of R-Modules and of R-Algebras
27(4)
1.9.1 Restriction and extension of scalars
29(1)
1.9.2 Tensor product of algebras
30(1)
1.10 Graded Rings and Modules, Homogeneous Ideals
31(11)
1.10.1 Homogeneous polynomials
35(6)
1.10.2 Graded modules and graded morphisms
41(1)
1.11 Localization
42(6)
1.11.1 Local rings and localization
46(2)
1.12 Krull-Dimension of a Ring
48(7)
Exercises
52(3)
2 Algebraic Afflne Sets
55(24)
2.1 Algebraic Affine Sets and Ideals
55(15)
2.2 Hilbert "Nullstellensatz"
70(4)
2.3 Some Consequences of Hilbert "Nullstellensatz" and of Elimination Theory
74(5)
2.3.1 Study's principle
74(1)
2.3.2 Intersections of affine plane curves
75(1)
Exercises
76(3)
3 Algebraic Projective Sets
79(30)
3.1 Algebraic Projective Sets
80(3)
3.2 Homogeneous "Hilbert Nullstellensatz"
83(3)
3.3 Fundamental Examples and Remarks
86(23)
3.3.1 Points
86(1)
3.3.2 Coordinate linear subspaces
87(1)
3.3.3 Hyperplanes and the dual projective space
87(1)
3.3.4 Fundamental affine open sets (or affine charts) of Fn
87(2)
3.3.5 Projective closure of affine sets
89(2)
3.3.6 Projective subspaces and their ideals
91(3)
3.3.7 Projective and affine subspaces
94(1)
3.3.8 Homographies, projectivities and affinities
95(4)
3.3.9 Projective cones
99(1)
3.3.10 Projective hypersurfaces and projective closure of affine hypersurfaces
99(1)
3.3.11 Proper closed subsets of P2
100(1)
3.3.12 Affine and projective twisted cubics
101(5)
Exercises
106(3)
4 Topological Properties and Algebraic Varieties
109(16)
4.1 Irreducible Topological Spaces
109(6)
4.1.1 Coordinate rings, ideals and irreducibility
112(2)
4.1.2 Algebraic varieties
114(1)
4.2 Noetherian Spaces: Irreducible Components
115(3)
4.3 Combinatorial Dimension
118(7)
Exercises
123(2)
5 Regular and Rational Functions on Algebraic Varieties
125(24)
5.1 Basics on Sheaves
125(2)
5.2 Regular Functions
127(3)
5.3 Rational Functions
130(19)
5.3.1 Consequences of the fundamental theorem on regular and rational functions
141(2)
5.3.2 Examples
143(4)
Exercises
147(2)
6 Morphisms of Algebraic Varieties
149(30)
6.1 Morphisms
149(2)
6.2 Morphisms with (Quasi) Affine Target
151(9)
6.3 Morphisms with (Quasi) Projective Target
160(4)
6.4 Local Properties of Morphisms: Affine Open Coverings of an Algebraic Variety
164(2)
6.5 Veronese Morphism: Divisors and Linear Systems
166(13)
6.5.1 Veronese morphism and consequences
171(3)
6.5.2 Divisors and linear systems
174(3)
Exercises
177(2)
7 Products of Algebraic Varieties
179(16)
7.1 Products of Affine Varieties
179(3)
7.2 Products of Projective Varieties
182(6)
7.2.1 Segre morphism and the product of projective spaces
184(2)
7.2.2 Products of projective varieties
186(2)
7.3 Products of Algebraic Varieties
188(1)
7.4 Products of Morphisms
189(2)
7.5 Diagonals, Graph of a Morphism and Fiber-Products
191(4)
Exercises
193(2)
8 Rational Maps of Algebraic Varieties
195(22)
8.1 Rational and Birational Maps
195(10)
8.1.1 Some properties and some examples of (bi)rational maps
199(6)
8.2 Unirational and Rational Varieties
205(12)
8.2.1 Stereographic projection of a rank-four quadric surface
206(2)
8.2.2 Monoids
208(1)
8.2.3 Blow-up of Pn at a point
209(3)
8.2.4 Blow-ups and resolution of singularities
212(3)
Exercises
215(2)
9 Completeness of Projective Varieties
217(8)
9.1 Complete Algebraic Varieties
217(2)
9.2 The Main Theorem of Elimination Theory
219(6)
9.2.1 Consequences of the main theorem of elimination theory
221(1)
Exercises
222(3)
10 Dimension of Algebraic Varieties
225(16)
10.1 Dimension of an Algebraic Variety
225(5)
10.2 Comparison on Various Definitions of "Dimension"
230(2)
10.3 Dimension and Intersections
232(3)
10.4 Complete Intersections
235(6)
Exercises
239(2)
11 Fiber-Dimension: Semicontinuity
241(8)
11.1 Fibers of a Dominant Morphism
241(3)
11.2 Semicontinuity
244(5)
Exercises
246(3)
12 Tangent Spaces: Smoothness of Algebraic Varieties
249(16)
12.1 Tangent Space at a Point of an Affine Variety: Smoothness
249(4)
12.2 Tangent Space at a Point of a Projective Variety: Smoothness
253(2)
12.3 Zariski Tangent Space of an Algebraic Variety: Intrinsic Definition of Smoothness
255(10)
Exercises
263(2)
Solutions to Exercises 265(30)
Bibliography 295(2)
Index 297