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A Royal Road to Algebraic Geometry / by Audun Holme
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: SpringerLink BücherPublisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2012Description: Online-Ressource (XIII, 364p. 27 illus, digital)ISBN:- 9783642192258
- 516
- 516.35
- QA564-609
- QA564
Contents:
Summary: Part I Curves: 1 Affine and Projective Space -- 2 Curves in A2 k and in P2 -- 3 Higher Geometry in the Projective Plane -- 4 Plane Curves and Algebra -- 5 Projective Varieties in PNk -- Part II Introduction to Grothendieck’s Theory of Schemes: 6 Categories and Functors -- 7 Constructions and Representable Functors -- 8 Abelian Categories -- 9 The Concept of Spec(A) -- 10 The Category of Schemes -- 11 Properties of Morphisms of Schemes -- 12 Modules, Algebras and Bundles on a Scheme -- 13 More Properties of Morphisms, Scheme Theoretic Image and the “Sorite” -- 14 Projective Schemes and Bundles -- 15 Further Properties of Morphisms -- 16 Conormal Sheaf and Projective Bundles -- 17 Cohomology Theory on Schemes -- 18 Intersection Theory -- 19 Characteristic Classes in Algebraic Geometry -- 20 The Riemann-Roch Theorem -- 21 Some Basic constructions in the category of projective kvarieties -- 22 More on Duality -- References -- Index.Summary: This book is about modern algebraic geometry. The title A Royal Road to Algebraic Geometry is inspired by the famous anecdote about the king asking Euclid if there really existed no simpler way for learning geometry, than to read all of his work Elements. Euclid is said to have answered: “There is no royal road to geometry!” The book starts by explaining this enigmatic answer, the aim of the book being to argue that indeed, in some sense there is a royal road to algebraic geometry. From a point of departure in algebraic curves, the exposition moves on to the present shape of the field, culminating with Alexander Grothendieck’s theory of schemes. Contemporary homological tools are explained. The reader will follow a directed path leading up to the main elements of modern algebraic geometry. When the road is completed, the reader is empowered to start navigating in this immense field, and to open up the door to a wonderful field of research. The greatest scientific experience of a lifetime!PPN: PPN: 1651085676Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""A Royal Road to Algebraic Geometry""; ""Preface""; ""Acknowledgment""; ""Contents""; ""Part I: Curves""; ""Chapter 1: Af ne and Projective Space""; ""1.1 De nitions""; ""1.2 Algebraic Subsets and Coordinates""; ""1.3 Af ne and Projective Coordinate Systems""; ""1.4 The Theorem of Desargues""; ""1.5 Duality for P2k""; ""Chapter 2: Curves in A2k and in P2k""; ""2.1 Conic Sections""; ""2.2 Singular and Non-singular Points""; ""2.3 Conics in the Projective Plane""; ""2.4 The Cubic Curves in A2k""; ""2.5 Elliptic Integrals and the Elliptic Transcendentals""; ""2.6 More Curves in A2R""
""2.7 General Af ne Algebraic Curves""""2.8 Singularities and Multiplicities""; ""2.9 Tangency""; ""Chapter 3: Higher Geometry in the Projective Plane""; ""3.1 Projective Curves""; ""3.2 Projective Closure and Af ne Restriction""; ""3.3 Smooth and Singular Points on Af ne and Projective Curves""; ""3.4 The Tangent to a Projective Curve""; ""3.5 Projective Equivalence""; ""3.6 Asymptotes""; ""3.7 General Conchoids""; ""3.8 The Dual Curve""; ""Chapter 4: Plane Curves and Algebra""; ""4.1 Af ne and Homogeneous Coordinate Rings""; ""4.2 Multiplicity and the Local Rings""
""4.3 Intersection Multiplicities for Af ne Plane Curves""""4.4 Intersection Theory for Curves in P2k""; ""4.5 Valuations and Valuation Rings""; ""4.6 Linear Systems of Plane Curves""; ""4.7 Af ne Restriction and Projective Closure""; ""4.8 Bézout's Theorem""; ""4.9 Algebraic Derivatives and the Jacobian""; ""4.10 Simple Elimination Theory""; ""4.11 An Application: The Twisted Cubic Curve""; ""4.12 Points of In exion and the Hessian""; ""Chapter 5: Projective Varieties in PNk""; ""5.1 Subvarieties of PNk""; ""5.2 Projective Non-singular Curves""
""5.3 Divisors on a Projective Non-singular Curve""""5.4 Smoothness and Tangency in any Dimension""; ""5.5 Hilbert Polynomial and Projective Invariants""; ""5.6 Emmy Noether, Her Family and Their Fate""; ""5.7 The Riemann-Roch Theorem for Non-singular Curves""; ""Part II: Introduction to Grothendieck's Theory of Schemes""; ""Chapter 6: Categories and Functors""; ""6.1 Objects and Morphisms""; ""6.2 Examples of Categories""; ""6.3 The Dual Category""; ""6.4 The Topology on a Topological Space Viewed as a Category""; ""6.5 Monomorphisms and Epimorphisms""; ""6.6 Isomorphisms""
""6.7 Covariant and Contravariant Functors""""6.8 Forgetful Functors""; ""6.9 The Category of Functors Fun(C,D)""; ""6.10 Functors of Several Variables""; ""6.11 Isomorphic and Equivalent Categories""; ""6.12 When are two Functors Isomorphic?""; ""6.13 Left and Right Adjoint Functors""; ""6.14 Representable Functors""; ""6.15 Representable Functors and Universal Properties: Yoneda's Lemma""; ""Chapter 7: Constructions and Representable Functors""; ""7.1 Products and Coproducts""; ""7.2 Fibered Products and Coproducts""; ""Chapter 8: Abelian Categories""; ""8.1 De nitions""
""8.2 Product and Coproduct in the Non Abelian Category of Commutative Rings""
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