Algebraic Geometry: A First Course by Joe Harris

[Greg Bernhardt]

• Author: Joe Harris
• Title: Algebraic Geometry: A First Course
• Amazon Link: https://www.amazon.com/dp/144193099X/?tag=pfamazon01-20
• Prerequisities:

Table of Contents:

[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]

 

[PKDfan]

The link sends me to a book about chemistry ("Elementary Principles of Chemical Processes").

 

[Greg Bernhardt]

The link sends me to a book about chemistry ("Elementary Principles of Chemical Processes").

fixed thanks!

 

[mathwonk]

this nice book by Joe is an attempt to make algebraic geometry more concrete than the usual super formal treatments and he succeeds. You will learn a lot here that you can actually use in practice to make calculations. Notice he found ti helpful to use this hands on approach even at elite schools like Brown and Harvard where he taught successfully from it. You should definitely read it with pencil and paper at hand.

 

[rezeew]

I find Algebraic Geometry: A First Course by Joe Harris to be a comprehensive and well-written text on the topic. The author's clear explanations and numerous examples make it accessible to readers with varying levels of mathematical background.

One of the strengths of this book is its organization. The table of contents provides a clear roadmap for readers to follow, starting with basic concepts such as varieties and maps and gradually building up to more advanced topics such as group actions and moduli spaces. The inclusion of exercises and hints for selected exercises also makes it a useful resource for self-study.

I appreciate the author's emphasis on providing both geometric and algebraic perspectives on the subject. This not only helps readers develop a deeper understanding of the material, but also highlights the connections between algebraic geometry and other areas of mathematics.

Additionally, the book covers a wide range of topics, including rational maps, determinantal varieties, and quadratic forms, making it a valuable reference for researchers in the field. Overall, I highly recommend Algebraic Geometry: A First Course to anyone interested in learning about this fascinating and important branch of mathematics.