- #1
- 19,522
- 10,250
- Author: Joe Harris
- Title: Algebraic Geometry: A First Course
- Amazon Link: https://www.amazon.com/dp/144193099X/?tag=pfamazon01-20
- Prerequisities:
Table of Contents:
Code:
[LIST]
[*] Preface
[*] Acknowledgments
[*] Using This Book
[*] Examples of Varieties and Maps
[LIST]
[*] Affine and Projective Varieties
[LIST]
[*] A Note About Our Field
[*] Affine Space and Affine Varieties
[*] Projective Space and Projective Varieties
[LIST]
[*] Linear Spaces
[*] Finite Sets
[*] Hypersurfaces
[*] Analytic Subvarieties and Submanifolds
[*] The Twisted Cubic
[*] Rational Normal Curves
[*] Determinantal Representation of the Rational Normal Curve
[*] Another Parametrization of the Rational Normal Curve
[*] The Family of Plane Conics
[*] A Synthetic Construction of the Rational Normal Curve
[*] Other Rational Curves
[*] Varieties Defined over Subfields of [itex]K[/itex]
[/LIST]
[*] A Note on Dimension, Smoothness, and Degree
[/LIST]
[*] Regular Functions and Maps
[LIST]
[*] The Zariski Topology
[*] Regular Functions on an Affine Variety
[*] Projective Varieties
[*] Regular Maps
[LIST]
[*] The Veronese Map
[*] Determinantal Representation of Veronese Varieties
[*] Subvarieties of Veronese Varieties
[*] The Segre Maps
[*] Subvarieties of Segre Varieties
[*] Products of Varieties
[*] Graphs
[*] Fiber Products
[*] Combinations of Veronese and Segre Maps
[/LIST]
[/LIST]
[*] Cones, Projections, and More About Products
[LIST]
[*] Cones
[*] Quadrics
[*] Projections
[*] More Cones
[*] More Projections
[*] Constructible Sets
[/LIST]
[*] Families and Parameter Spaces
[LIST]
[*] Families of Varieties
[*] The Universal Hyperplane
[*] The Universal Hyperplane Section
[*] Parameter Spaces of Hypersurfaces
[*] Universal Families of Hypersurfaces
[*] A Family of Lines
[/LIST]
[*] Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz
[LIST]
[*] Generating Ideals
[*] Ideals of Projective Varieties
[*] Irreducible Varieties and Irreducible Decomposition
[*] General Objects
[LIST]
[*] General Projections
[*] General Twisted Cubics
[*] Double Point Loci
[/LIST]
[*] A Little Algebra
[*] Restatements and Corollaries
[/LIST]
[*] Grassmannians and Related Varieties
[LIST]
[*] Grassmannians
[*] Subvarieties of Grassmannians
[*] The Grassmannian [itex]\mathbb{G}(1, 3)[/itex]
[*] An Analog of the Veronese Map
[*] Incidence Correspondences
[*] Varieties of Incident Planes
[*] The Join of Two Varieties
[*] Fano Varieties
[/LIST]
[*] Rational Functions and Rational Maps
[LIST]
[*] Rational Functions
[*] Rational Maps
[*] Graphs of Rational Maps
[*] Birational Isomorphism
[LIST]
[*] The Quadric Surface
[*] Hypersurfaces
[/LIST]
[*] Degree of a Rational Map
[*] Blow-Ups
[LIST]
[*] Blowing Up Points
[*] Blowing Up Subvarieties
[*] The Quadric Surface Again
[*] The Cubic Scroll in [itex]\mathbb{P}^4[/itex]
[/LIST]
[*] Unirationality
[/LIST]
[*] More Examples
[LIST]
[*] The Join of Two Varieties
[*] The Secant Plane Map
[*] Secant Varieties
[*] Trisecant Lines, etc.
[*] Joins of Corresponding Points
[*] Rational Normal Scrolls
[*] Higher-Dimensional Scrolls
[*] More Incidence Correspondences
[*] Flag Manifolds
[*] More Joins and Intersections
[*] Quadrics of Rank 4
[*] Rational Normal Scrolls II
[/LIST]
[*] Determinantal Varieties
[LIST]
[*] Generic Determinantal Varieties
[LIST]
[*] Segre Varieties
[*] Secant Varieties of Segre Varieties
[/LIST]
[*] Linear Determinantal Varieties in General
[LIST]
[*] Rational Normal Curves
[*] Secant Varieties to Rational Normal Curves
[*] Rational Normal Scrolls III
[*] Rational Normal Scrolls IV
[/LIST]
[*] More General Determinantal Varieties
[*] Symmetric and Skew-Symmetric Determinantal Varieties
[LIST]
[*] Fano Varieties of Determinantal Varieties
[/LIST]
[/LIST]
[*] Algebraic Groups
[LIST]
[*] The General Linear Group [itex]GL_nK[/itex]
[*] The Orthogonal Group [itex]SO_nK[/itex]
[*] The Symplectic Group [itex]Sp_{2n}K[/itex]
[*] Group Actions
[LIST]
[*] [itex]PGL_{n+1}K[/itex] acts on [itex]\mathbb{P}^n[/itex]
[*] [itex]PGL_2K[/itex] acts on [itex]\mathbb{P}^2[/itex]
[*] [itex]PGL_2K[/itex] acts on [itex]\mathbb{P}^3[/itex]
[*] [itex]PGL_2K[/itex] acts on [itex]\mathbb{P}^n[/itex]
[*] [itex]PGL_3K[/itex] acts on [itex]\mathbb{P}^5[/itex]
[*] [itex]PGL_3K[/itex] acts on [itex]\mathbb{P}^9[/itex]
[*] [itex]PO_nK[/itex] acts on [itex]\mathbb{P}^{n-1}[/itex] (automorphisms of the Grassmannian)
[*] [itex]PGL_n(K)[/itex] acts on [itex]\mathbb{P}\left(\bigwedge^k K^n\right)[/itex]
[/LIST]
[*] Quotients
[*] Quotients of Affine Varieties by Finite Groups
[LIST]
[*] Quotients of Affine Space
[*] Symmetric Products
[/LIST]
[*] Quotients of Projective Varieties by Finite Groups
[LIST]
[*] Weighted Projective Spaces
[/LIST]
[/LIST]
[/LIST]
[*] Attributes of Varieties
[LIST]
[*] Definitions of Dimension and Elementary Examples
[LIST]
[*] Hypersurfaces
[*] Complete Intersections
[*] Immediate Examples
[LIST]
[*] The Universal [itex]k[/itex]-Plane
[*] Varieties of Incident Planes
[*] Secant Varieties
[*] Secant Varieties in General
[*] Joins of Varieties
[*] Flag Manifolds
[*] (Some) Schubert Varieties
[/LIST]
[/LIST]
[*] More Dimension Computations
[LIST]
[*] Determinantal Varieties
[*] Fano Varieties
[*] Parameter Spaces of Twisted Cubics
[LIST]
[*] Twisted Cubics
[*] Cubics on a General Surface
[*] Complete Intersections
[*] Curves of Type [itex](a, b)[/itex] on a Quadric
[*] Determinantal Varieties
[/LIST]
[*] Group Actions
[LIST]
[*] [itex]GL(V)[/itex] Acts on [itex]Sym^dV[/itex] and [itex]\bigwedge^k V[/itex]
[*] [itex]PGL_{n+1}K[/itex] Acts on [itex](\mathbb{P}^n)^l[/itex] and [itex]\mathbb{G}(k,n)^l[/itex]
[/LIST]
[/LIST]
[*] Hilbert Polynomials
[LIST]
[*] Hilbert Functions and Polynomials
[LIST]
[*] Hilbert Function of the Rational Normal Curve
[*] Hilbert Function of the Veronese Variety
[*] Hilbert Polynomials of Curves
[/LIST]
[*] Syzygies
[LIST]
[*] Three Points in [itex]\mathbb{P}^2[/itex]
[*] Four Points in [itex]\mathbb{P}^2[/itex]
[*] Complete Intersections: Koszul Complexes
[/LIST]
[/LIST]
[*] Smoothness and Tangent Spaces
[LIST]
[*] The Zariski Tangent Space to a Variety
[*] A Local Criterion for Isomorphism
[*] Projective Tangent Spaces
[*] Determinantal Varieties
[/LIST]
[*] Gauss Maps, Tangential and Dual Varieties
[LIST]
[*] A Note About Characteristic
[*] Gauss Maps
[*] Tangential Varieties
[*] The Variety of Tangent Lines
[*] Joins of Intersecting Varieties
[*] The Locus of Bitangent Lines
[*] Dual Varieties
[/LIST]
[*] Tangent Spaces to Grassmannians
[LIST]
[*] Tangent Spaces to Grassmannians
[*] Tangent Spaces to Incidence Correspondences
[*] Varieties of Incident Planes
[*] The Variety of Secant Lines
[*] Varieties Swept out by Linear Spaces
[*] The Resolution of the Generic Determinantal Variety
[*] Tangent Spaces to Dual Varieties
[*] Tangent Spaces to Fano Varieties
[/LIST]
[*] Further Topics Involving Smoothness and Tangent Spaces
[LIST]
[*] Gauss Maps on Curves
[*] Osculating Planes and Associated Maps
[*] The Second Fundamental Form
[*] The Locus of Tangent Lines to a Variety
[*] Bertini's Theorem
[*] Blow-ups, Nash Blow-ups, and the Resolution of Singularities
[*] Subadditivity of Codimensions of Intersections
[/LIST]
[*] Degree
[LIST]
[*] Bézout's Theorem
[LIST]
[*] The Rational Normal Curves
[/LIST]
[*] More Examples of Degrees
[LIST]
[*] Veronese Varieties
[*] Segre Varieties
[*] Degrees of Cones and Projections
[*] Joins of Varieties
[*] Unirationality of Cubic Hypersurfaces
[/LIST]
[/LIST]
[*] Further Examples and Applications of Degree
[LIST]
[*] Multidegree of a Subvariety of a Product
[*] Projective Degree of a Map
[*] Joins of Corresponding Points
[*] Varieties of Minimal Degree
[*] Degrees of Determinantal Varieties
[*] Degrees of Varieties Swept out by Linear Spaces
[*] Degrees of Some Grassmannians
[*] Harnack's Theorem
[/LIST]
[*] Singular Points and Tangent Cones
[LIST]
[*] Tangent Cones
[LIST]
[*] Tangent Cones to Determinantal Varieties
[/LIST]
[*] Multiplicity
[*] Examples of Singularities
[*] Resolution of Singularities for Curves
[/LIST]
[*] Parameter Spaces and Moduli Spaces
[LIST]
[*] Parameter Spaces
[*] Chow Varieties
[*] Hilbert Varieties
[*] Curves of Degree 2
[*] Moduti Spaces
[LIST]
[*] Plane Cubics
[/LIST]
[/LIST]
[*] Quadrics
[LIST]
[*] Generalities about Quadrics
[*]Tangent Spaces to Quadrics
[*] Plane Conics
[*] Quadric Surfaces
[*] Quadrics in [itex]\mathbb{P}^n[/itex]
[*] Linear Spaces on Quadrics
[LIST]
[*] Lines on Quadrics
[*] Planes on Four-Dimensional Quadrics
[*] Fano Varieties of Quadrics in General
[/LIST]
[*] Families of Quadrics
[LIST]
[*] The Variety of Quadrics in [itex]\mathbb{P}^1[/itex]
[*] The Variety of Quadrics in [itex]\mathbb{P}^2[/itex]
[*] Complete Conics
[*] Quadrics in [itex]\mathbb{P}^n[/itex]
[/LIST]
[*] Pencils of Quadrics
[/LIST]
[/LIST]
[*] Hints for Selected Exercises
[*] References
[*] Index
[/LIST]
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