Algebraic Geometry: A First Course (Graduate Texts in Mathematics, 133)

✍ Scribed by Joe Harris

• Publisher: Springer
• Year: 1992
• Tongue: English
• Leaves: 349
• Edition: Corrected
• Category: Library

✦ Synopsis

This book is based on one-semester courses given at Harvard in 1984, at Brown in 1985, and at Harvard in 1988. It is intended to be, as the title suggests, a first introduction to the subject. Even so, a few words are in order about the purposes of the book. Algebraic geometry has developed tremendously over the last century. During the 19th century, the subject was practiced on a relatively concrete, down-to-earth level; the main objects of study were projective varieties, and the techniques for the most part were grounded in geometric constructions. This approach flourished during the middle of the century and reached its culmination in the work of the Italian school around the end of the 19th and the beginning of the 20th centuries. Ultimately, the subject was pushed beyond the limits of its foundations: by the end of its period the Italian school had progressed to the point where the language and techniques of the subject could no longer serve to express or carry out the ideas of its best practitioners.

✦ Table of Contents

Cover
Title Page
Copyright Page
Dedication
Preface
Acknowledgments
Using This Book
Contents
Part I: Examples of Varieties and Maps
Lecture 1. Affine and Projective Varieties
A Note About Our Field
Affine Space and Affine Varieties
Projective Space and Projective Varieties
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 K
A Note on Dimension, Smoothness, and Degree
Lecture 2. Regular Functions and Maps
The Zariski Topology
Regular Functions on an Affine Variety
Projective Varieties
Regular Maps
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
Lecture 3. Cones, Projections, and More About Products
Cones
Quadrics
Projections
More Cones
More Projections
Constructible Sets
Lecture 4. Families and Parameter Spaces
Families of Varieties
The Universal Hyperplane
The Universal Hyperplane Section
Parameter Spaces of Hypersurfaces
Universal Families of Hypersurfaces
A Family of Lines
Lecture 5. Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz
Generating Ideals
Ideals of Projective Varieties
Irreducible Varieties and Irreducible Decomposition
General Objects
General Projections
General Twisted Cubics
Double Point Loci
A Little Algebra
Restatements and Corollaries
Lecture 6. Grassmannians and Related Varieties
Grassmannians
Subvarieties of Grassmannians
The Grassmannian G(1, 3)
An Analog of the Veronese Map
Incidence Correspondences
Varieties of Incident Planes
The Join of Two Varieties
Fano Varieties
Lecture 7. Rational Functions and Rational Maps
Rational Functions
Rational Maps
Graphs of Rational Maps
Birational Isomorphism
The Quadric Surface
Hypersurfaces
Degree of a Rational Map
Blow-Ups
Blowing Up Points
Blowing up Subvarieties
The Quadric Surface Again
The Cubic Scroll in P^4
Unirationality
Lecture 8. More Examples
The Join of Two Varieties
The Secant Plane Maps
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
Lecture 9. Determinantal Varieties
Generic Determinantal Varieties
Segre Varieties
Secant Varieties of Segre Varieties
Linear Determinantal Varieties in General
Rational Normal Curves
Secant Varieties to Rational Normal Curves
Rational Normal Scrolls III
Rational Normal Scrolls IV
More General Determinantal Varieties
Symmetric and Skew-Symmetric Determinantal Varieties
Fano Varieties of Determinantal Varieties
Lecture 10. Algebraic Groups
The General Linear Group GL n K

The Symplectic Group Sp 2n K
Group Actions
PGL n+1 K Acts on P^n
PGL 2 K Acts on P^2
PGL 2 K acts on P^3
PGL 2 K Acts on P^n (and on P^4 in particular)
PGL 3 K Acts on P^5
PGL 3 K Acts on P^9
PO n K Acts on P^n–1 (automorphisms of the Grassmannian)

Quotients
Quotients of Affine Varieties by Finite Groups
Quotients of Affine Space
Symmetric Products
Quotients of Projective Varieties by Finite Groups
Weighted Projective Spaces
Part II: Attributes of Varieties
Lecture 12. More Dimension Computations
Determinantal Varieties
Fano Varieties
Parameter Spaces of Twisted Cubics
Twisted Cubics
Twisted Cubics on a General Surface
Complete Intersections
Curves of Type (a, b) on a Quadric
Determinantal Varieties
Group Actions

PGL n+1 K Acts on (P^n)^l and G(k, n)^l
Lecture 13. Hilbert Polynomials
Hilbert Functions and Polynomials
Hilbert Function of the Rational Normal Curve
Hilbert Function of the Veronese Variety
Hilbert Polynomials of Curves (for those familiar with the Riemann–Roch theorem for curves)
Syzygies
Three Points in P^2
Four Points in P^2
Complete Intersections: Koszul Complexes
Lecture 14. Smoothness and Tangent Spaces
The Zariski Tangent Space to a Variety
A Local Criterion for Isomorphism
Projective Tangent Spaces
Determinantal Varieties
Lecture 15. Gauss Maps, Tangential and Dual Varieties
A Note About Characteristic
Gauss Maps
Tangential Varieties
The Variety of Tangent Lines
Joins of Intersecting Varieties
The Locus of Bitangent Lines
Dual Varieties
Lecture 16. Tangent Spaces to Grassmannians
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
Lecture 17. Further Topics Involving Smoothness and Tangent Spaces
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 Codimension of Intersections
Lecture 18. Degree
Bezout's Theorem
The Rational Normal Curve
More Examples of Degrees
Veronese Varieties
Segre Varieties
Degrees of Cones and Projections
Joins of Varieties
Unirationality of Cubic Hypersurfaces
Lecture 19. Further Examples and Applications of Degree
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
Lecture 20. Singular Points and Tangent Cones
Tangent Cones
Tangent Cones to Determinantal Varieties
Multiplicity
Examples of Singularities
Resolution of Singularities for Curves
Lecture 21. Parameter Spaces and Moduli Spaces
Parameter Spaces
Chow Varieties
Hilbert Varieties
Curves of Degree 2
Moduli Spaces
Plane Cubics
Lecture 22. Quadrics
Generalities about Quadrics
Tangent Spaces to Quadrics
Plane Conics
Quadric Surfaces
Quadrics in P^n
Linear Spaces on Quadrics
Lines on Quadrics
Planes on Four-Dimensional Quadrics
Fano Varieties of Quadrics in General
Families of Quadrics
The Variety of Quadrics in P^1
The Variety of Quadrics in P^2
Complete Conics
Quadrics in P^n
Pencils of Quadrics
Lecture 11. Definitions of Dimension and Elementary Examples
Hypersurfaces
Complete Intersections
Immediate Examples
The Universal k-Plane
Varieties of Incident Planes
Secant Varieties
Secant Varieties in General
Joins of Varieties
Flag Manifolds
(Some) Schubert Cycles
Hints for Selected Exercises
References
Index