| I Basic Definitions I.1 Affine Schemes I.1.1 Schemes as Sets I.1.2 Schemes as Topological Spaces I.1.3 An Interlude on Sheaf Theory References for the Theory of Sheaves I.1.4 Schemes as Schemes (Structure Sheaves) I.2 Schemes in General I.2.1 Subschemes I.2.2 The Local Ring at a Point I.2.3 Morphisms I.2.4 The Gluing Construction Projective Space I.3 Relative Schemes I.3.1 Fibered Products I.3.2 The Category of S-Schemes I.3.3 Global Spec I.4 The Functor of Points II Examples II.1 Reduced Schemes over Algebraically Closed Fields II. 1.1 Affine Spaces II.1.2 Local Schemes II.2 Reduced Schemes over Non-Algebraically Closed Fields II.3 Nonreduced Schemes II.3.1 Double Points II.3.2 Multiple Points Degree and Multiplicity II.3.3 Embedded Points Primary Decomposition II.3.4 Flat Families of Schemes Limits Examples Flatness II.3.5 Multiple Lines II.4 Arithmetic Schemes II.4.1 Spec Z II.4.2 Spec of the Ring of Integers in a Number Field II.4.3 Affine Spaces over Spec Z II.4.4 A Conic over Spec Z II.4.5 Double Points in Al III Projective Schemes III.1 Attributes of Morphisms III.1.1 Finiteness Conditions III.1.2 Properness and Separation III.2 Proj of a Graded Ring III.2.1 The Construction of Proj S III.2.2 Closed Subschemes of Proj R III.2.3 Global Proj Proj of a Sheaf of Graded 0x-Algebras The Projectivization P(ε) of a Coherent Sheaf ε III.2.4 Tangent Spaces and Tangent Cones Affine and Projective Tangent Spaces Tangent Cones III.2.5 Morphisms to Projective Space III.2.6 Graded Modules and Sheaves III.2.7 Grassmannians III.2.8 Universal Hypersurfaces III.3 Invariants of Projective Schemes III.3.1 Hilbert Functions and Hilbert Polynomials 1II.3.2 Flatness Il： Families of Projective Schemes III.3.3 Free Resolutions III.3.4 Examples Points in the Plane Examples： Double Lines in General and in p3 III.3.5 BEzouts Theorem Multiplicity of Intersections III.3.6 Hilbert Series IV Classical Constructions IV.1 Flexes of Plane Curves IV.I.1 Definitions IV.1.2 Flexes on Singular Curves IV.1.3 Curves with Multiple Components IV.2 Blow-ups IV.2.1 Definitions and Constructions An Example： Blowing up the Plane Definition of Blow-ups in General The Blowup as Proj Blow-ups along Regular Subschemes IV.2.2 Some Classic Blow-Ups IV.2.3 Blow-ups along Nonreduced Schemes Blowing Up a Double Point Blowing Up Multiple Points The j-Function IV.2.4 Blow-ups of Arithmetic Schemes IV.2.5 Project： Quadric and Cubic Surfaces as Blow-ups IV.3 Fano schemes IV.3.1 Definitions IV.3.2 Lines on Quadrics Lines on a Smooth Quadric over an Algebraically Closed Field Lines on a Quadric Cone A Quadric Degenerating to Two Planes More Examples IV.3.3 Lines on Cubic Surfaces IV.4 Forms V Local Constructions V.1 Images V.I.1 The Image of a Morphism of Schemes V.1.2 Universal Formulas V.1.3 Fitting Ideals and Fitting Images Fitting Ideals Fitting Images V.2 Resultants V.2：l Definition of the Resultant V.2.2 Sylvesters Determinant V.3 Singular Schemes and Discriminants V.3.1 Definitions V.3.2 Discriminants V.3.3 Examples V.4 Dual Curves V.4.1 Definitions V.4.2 Duals of Singular Curves V.4.3 Curves with Multiple Components V.5 Double Point Loci VI Schemes and Functors VI.1 The Functor of Points VI.I.1 Open and Closed Subfunctors VI.1.2 K-Rational Points VI.1.3 Tangent Spaces to a Functor VI.1.4 Group Schemes VI.2 Characterization of a Space by its ~nctor of Points VI.2.1 Characterization of Schemes among Functors VI.2.2 Parameter Spaces The Hilbert Scheme Examples of Hilbert Schemes Variations on the Hilbert Scheme Construction. VI.2.3 Tangent Spaces to Schemes in Terms of Their Func tors of Points Tangent Spaces to Hilbert Schemes Tangent Spaces to Fano Schemes VI.2.4 Moduli Spaces References Index |
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