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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.

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• Preface(p. vii)
• Acknowledgments(p. ix)
• Using This Book(p. xi)
• Part I Examples of Varieties and Maps
• Lecture 1 Affine and Projective Varieties(p. 3)
• A Note About Our Field(p. 3)
• Affine Space and Affine Varieties(p. 3)
• Projective Space and Projective Varieties(p. 3)
• Linear Spaces(p. 5)
• Finite Sets(p. 6)
• Hypersurfaces(p. 8)
• Analytic Subvarieties and Submanifolds(p. 8)
• The Twisted Cubic(p. 9)
• Rational Normal Curves(p. 10)
• Determinantal Representation of the Rational Normal Curve(p. 11)
• Another Parametrization of the Rational Normal Curve(p. 11)
• The Family of Plane Conics(p. 12)
• A Synthetic Construction of the Rational Normal Curve(p. 13)
• Other Rational Curves(p. 14)
• Varieties Defined over Subfields of K(p. 16)
• A Note on Dimension, Smoothness, and Degree(p. 16)
• Lecture 2 Regular Functions and Maps(p. 17)
• The Zariski Topology(p. 17)
• Regular Functions on an Affine Variety(p. 18)
• Projective Varieties(p. 20)
• Regular Maps(p. 21)
• The Veronese Map(p. 23)
• Determinantal Representation of Veronese Varieties(p. 24)
• Subvarieties of Veronese Varieties(p. 24)
• The Segre Maps(p. 25)
• Subvarieties of Segre Varieties(p. 27)
• Products of Varieties(p. 28)
• Graphs(p. 29)
• Fiber Products(p. 30)
• Combinations of Veronese and Segre Maps(p. 30)
• Lecture 3 Cones, Projections, and More About Products(p. 32)
• Cones(p. 32)
• Quadrics(p. 33)
• Projections(p. 34)
• More Cones(p. 37)
• More Projections(p. 38)
• Constructible Sets(p. 39)
• Lecture 4 Families and Parameter Spaces(p. 41)
• Families of Varieties(p. 41)
• The Universal Hyperplane(p. 42)
• The Universal Hyperplane Section(p. 43)
• Parameter Spaces of Hypersurfaces(p. 44)
• Universal Families of Hypersurfaces(p. 45)
• A Family of Lines(p. 47)
• Lecture 5 Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz(p. 48)
• Generating Ideals(p. 48)
• Ideals of Projective Varieties(p. 50)
• Irreducible Varieties and Irreducible Decomposition(p. 51)
• General Objects(p. 53)
• General Projections(p. 54)
• General Twisted Cubics(p. 55)
• Double Point Loci(p. 56)
• A Little Algebra(p. 57)
• Restatements and Corollaries(p. 60)
• Lecture 6 Grassmannians and Related Varieties(p. 63)
• Grassmannians(p. 63)
• Subvarieties of Grassmannians(p. 66)
• The Grassmannian G(1, 3)(p. 67)
• An Analog of the Veronese Map(p. 68)
• Incidence Correspondences(p. 68)
• Varieties of Incident Planes(p. 69)
• The Join of Two Varieties(p. 70)
• Fano Varieties(p. 70)
• Lecture 7 Rational Functions and Rational Maps(p. 72)
• Rational Functions(p. 72)
• Rational Maps(p. 73)
• Graphs of Rational Maps(p. 75)
• Birational Isomorphism(p. 77)
• The Quadric Surface(p. 78)
• Hypersurfaces(p. 79)
• Degree of a Rational Map(p. 79)
• Blow-Ups(p. 80)
• Blowing Up Points(p. 81)
• Blowing Up Subvarieties(p. 82)
• The Quadric Surface Again(p. 84)
• The Cubic Scroll in P[superscript 4](p. 85)
• Unirationality(p. 87)
• Lecture 8 More Examples(p. 88)
• The Join of Two Varieties(p. 88)
• The Secant Plane Map(p. 89)
• Secant Varieties(p. 90)
• Trisecant Lines, etc.(p. 90)
• Joins of Corresponding Points(p. 91)
• Rational Normal Scrolls(p. 92)
• Higher-Dimensional Scrolls(p. 93)
• More Incidence Correspondences(p. 94)
• Flag Manifolds(p. 95)
• More Joins and Intersections(p. 95)
• Quadrics of Rank 4(p. 96)
• Rational Normal Scrolls II(p. 97)
• Lecture 9 Determinantal Varieties(p. 98)
• Generic Determinantal Varieties(p. 98)
• Segre Varieties(p. 98)
• Secant Varieties of Segre Varieties(p. 99)
• Linear Determinantal Varieties in General(p. 99)
• Rational Normal Curves(p. 100)
• Secant Varieties to Rational Normal Curves(p. 103)
• Rational Normal Scrolls III(p. 105)
• Rational Normal Scrolls IV(p. 109)
• More General Determinantal Varieties(p. 111)
• Symmetric and Skew-Symmetric Determinantal Varieties(p. 112)
• Fano Varieties of Determinantal Varieties(p. 112)
• Lecture 10 Algebraic Groups(p. 114)
• The General Linear Group GL[subscript n]K(p. 114)
• The Orthogonal Group SO[subscript n]K(p. 115)
• The Symplectic Group Sp[subscript 2n]K(p. 116)
• Group Actions(p. 116)
• PGL[subscript n+1]K acts on P[superscript n](p. 116)
• PGL[subscript 2]K Acts on P[superscript 2](p. 117)
• PGL[subscript 2]K Acts on P[superscript 3](p. 118)
• PGL[subscript 2]K Acts on P[superscript n](p. 119)
• PGL[subscript 3]K Acts on P[superscript 5](p. 120)
• PGL[subscript 3]K Acts on P[superscript 9](p. 121)
• PO[subscript n]K Acts on P[superscript n-1] (automorphisms of the Grassmannian)(p. 122)
• PGL[subscript n]K Acts on P([logical and superscript k]K[superscript n])(p. 122)
• Quotients(p. 123)
• Quotients of Affine Varieties by Finite Groups(p. 124)
• Quotients of Affine Space(p. 125)
• Symmetric Products(p. 126)
• Quotients of Projective Varieties by Finite Groups(p. 126)
• Weighted Projective Spaces(p. 127)
• Part II Attributes of Varieties
• Lecture 11 Definitions of Dimension and Elementary Examples(p. 133)
• Hypersurfaces(p. 136)
• Complete Intersections(p. 136)
• Immediate Examples(p. 138)
• The Universal k-Plane(p. 142)
• Varieties of Incident Planes(p. 142)
• Secant Varieties(p. 143)
• Secant Varieties in General(p. 146)
• Joins of Varieties(p. 148)
• Flag Manifolds(p. 148)
• (Some) Schubert Varieties(p. 149)
• Lecture 12 More Dimension Computations(p. 151)
• Determinantal Varieties(p. 151)
• Fano Varieties(p. 152)
• Parameter Spaces of Twisted Cubics(p. 155)
• Twisted Cubics(p. 155)
• Twisted Cubics on a General Surface(p. 156)
• Complete Intersections(p. 157)
• Curves of Type (a, b) on a Quadric(p. 158)
• Determinantal Varieties(p. 159)
• Group Actions(p. 161)
• GL (V) Acts on Sym[superscript d]V and [logical and superscript k]V(p. 161)
• PGL[subscript n+1]K Acts on (P[superscript n])[superscript l] and G(k, n)[superscript l](p. 161)
• Lecture 13 Hilbert Polynomials(p. 163)
• Hilbert Functions and Polynomials(p. 163)
• Hilbert Function of the Rational Normal Curve(p. 166)
• Hilbert Function of the Veronese Variety(p. 166)
• Hilbert Polynomials of Curves(p. 166)
• Syzygies(p. 168)
• Three Points in P[superscript 2](p. 170)
• Four Points in P[superscript 2](p. 171)
• Complete Intersections: Koszul Complexes(p. 172)
• Lecture 14 Smoothness and Tangent Spaces(p. 174)
• The Zariski Tangent Space to a Variety(p. 174)
• A Local Criterion for Isomorphism(p. 177)
• Projective Tangent Spaces(p. 181)
• Determinantal Varieties(p. 184)
• Lecture 15 Gauss Maps, Tangential and Dual Varieties(p. 186)
• A Note About Characteristic(p. 186)
• Gauss Maps(p. 188)
• Tangential Varieties(p. 189)
• The Variety of Tangent Lines(p. 190)
• Joins of Intersecting Varieties(p. 193)
• The Locus of Bitangent Lines(p. 195)
• Dual Varieties(p. 196)
• Lecture 16 Tangent Spaces to Grassmannians(p. 200)
• Tangent Spaces to Grassmannians(p. 200)
• Tangent Spaces to Incidence Correspondences(p. 202)
• Varieties of Incident Planes(p. 203)
• The Variety of Secant Lines(p. 204)
• Varieties Swept out by Linear Spaces(p. 204)
• The Resolution of the Generic Determinantal Variety(p. 206)
• Tangent Spaces to Dual Varieties(p. 208)
• Tangent Spaces to Fano Varieties(p. 209)
• Lecture 17 Further Topics Involving Smoothness and Tangent Spaces(p. 211)
• Gauss Maps on Curves(p. 211)
• Osculating Planes and Associated Maps(p. 213)
• The Second Fundamental Form(p. 214)
• The Locus of Tangent Lines to a Variety(p. 215)
• Bertini's Theorem(p. 216)
• Blow-ups, Nash Blow-ups, and the Resolution of Singularities(p. 219)
• Subadditivity of Codimensions of Intersections(p. 222)
• Lecture 18 Degree(p. 224)
• Bezout's Theorem(p. 227)
• The Rational Normal Curves(p. 229)
• More Examples of Degrees(p. 231)
• Veronese Varieties(p. 231)
• Segre Varieties(p. 233)
• Degrees of Cones and Projections(p. 234)
• Joins of Varieties(p. 235)
• Unirationality of Cubic Hypersurfaces(p. 237)
• Lecture 19 Further Examples and Applications of Degree(p. 239)
• Multidegree of a Subvariety of a Product(p. 239)
• Projective Degree of a Map(p. 240)
• Joins of Corresponding Points(p. 241)
• Varieties of Minimal Degree(p. 242)
• Degrees of Determinantal Varieties(p. 243)
• Degrees of Varieties Swept out by Linear Spaces(p. 244)
• Degrees of Some Grassmannians(p. 245)
• Harnack's Theorem(p. 247)
• Lecture 20 Singular Points and Tangent Cones(p. 251)
• Tangent Cones(p. 251)
• Tangent Cones to Determinantal Varieties(p. 256)
• Multiplicity(p. 258)
• Examples of Singularities(p. 260)
• Resolution of Singularities for Curves(p. 264)
• Lecture 21 Parameter Spaces and Moduli Spaces(p. 266)
• Parameter Spaces(p. 266)
• Chow Varieties(p. 268)
• Hilbert Varieties(p. 273)
• Curves of Degree 2(p. 275)
• Moduli Spaces(p. 278)
• Plane Cubics(p. 279)
• Lecture 22 Quadrics(p. 282)
• Generalities about Quadrics(p. 282)
• Tangent Spaces to Quadrics(p. 283)
• Plane Conics(p. 284)
• Quadric Surfaces(p. 285)
• Quadrics in P[superscript n](p. 287)
• Linear Spaces on Quadrics(p. 289)
• Lines on Quadrics(p. 290)
• Planes on Four-Dimensional Quadrics(p. 291)
• Fano Varieties of Quadrics in General(p. 293)
• Families of Quadrics(p. 295)
• The Variety of Quadrics in P[superscript 1](p. 295)
• The Variety of Quadrics in P[superscript 2](p. 296)
• Complete Conics(p. 297)
• Quadrics in P[superscript n](p. 299)
• Pencils of Quadrics(p. 301)
• Hints for Selected Exercises(p. 308)
• References(p. 314)
• Index(p. 317)

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CHOICE Review

Nowadays, geometry in its many forms suffers neglect at the undergraduate level. The curriculum is still dominated by abstract algebra, real analysis, and general topology, reflecting mathematical fashions of 40 years ago. In the early part of this century projective geometry was indeed a standard undergraduate subject. Then, between 1930 and 1970, for the sake of rigor and generality, algebraic geometry received a radical new foundation at the hands of Weil, Zariski, Serre, Grothendieck, and others. The concrete language of classical algebraic geometry was rendered obsolete, though its results, suitably reformulated, remain valid and important. Unfortunately, the language of modern algebraic geometry is much too abstract to present to undergraduates. Although Harris says the basis of his text is the many topics that are "nearly as well understood classically as they are in modern language," the book is really a bridge between the two perspectives. Some of the most important passages show the breaking points of classical theory. Modern theorems are stated, often without proof, when they address classical concerns. Detailed examinations of geometrical notions such as dimension and degree reveal the pathologies that justify more abstract formulations. The book is arranged as 22 informal, relatively independent, lectures. The almost zoological character of classical geometry is reflected in the wealth of examples filling nearly the first half of the book, including many of higher dimensions as well as the usual curves and surfaces. Cox, Little, and O'Shea offer an even more elementary introduction to algebraic geometry from an algebraic perspective. Their emphasis is on the algebra of polynomials that students meet first in high school. In college linear algebra courses students learn to manipulate linear equations in many varibles. This book treats the more difficult problem of manipulating higher-degree equations in many variables. Many procedures described here were discovered only in the last 30 years. These procedures form the basis for computer algebra systems such as Maple, Mathematica, and REDUCE; the reader is encouraged to use these systems to explore the various concepts. One chapter explores significant applications to robotics and automatic theorem proving. No background in abstract algebra is presumed. Because a student meets rings, ideals, and even groups here in an especially concrete setting, a course based on this text would be an attractive alternative to a traditional abstract algebra course, leaving the student well prepared for more advanced study of algebraic geometry. Both of these books are highly recommended for all academic libraries. D. V. Feldman; University of New Hampshire