Algebraic geometry is the crown jewel of pure mathematics, the sacred ground where the structural rigidity of algebra fuses with the spatial intuition of geometry. For generations of mathematicians, stepping into this domain meant confronting a daunting wall of abstraction. "Introduction to Algebraic Geometry" book whether it is a classical doorway, a computational path, or a modern portal into scheme theory, serves as the ultimate bridge. It takes the familiar world of polynomial equations and transforms it into a vast landscape of geometric forms, proving that what we calculate and what we visualize are two sides of the exact same coin. At its heart, algebraic geometry asks a deceptively simple question: What do the solutions to a system of polynomial equations actually look like? An introductory textbook takes you on a journey from basic high school algebra to the frontiers of modern research. It begins with affine and projective varieties, mapping out how simple zero-sets of polynomials weave intricate shapes in multi-dimensional space. It gently transitions the reader from concrete coordinate spaces to the brilliant, intrinsic language of sheaves, schemes, and cohomology, unlocking the tools that revolutionized 20th-century mathematics. This volume offers a rapid, concise, and self-contained introductory approach to the algebraic aspects of the third method, the algebraico-geometric. The treatment assumes only familiarity with elementary algebra up to the level of Galois theory. Starting with an opening chapter on the general theory of places, the author advances to examinations of algebraic varieties, the absolute theory of varieties, and products, projections, and correspondences. Subsequent chapters explore normal varieties, divisors and linear systems, differential forms, the theory of simple points, and algebraic groups, concluding with a focus on the Riemann-Roch theorem. All the theorems of a general nature related to the foundations of the theory of algebraic groups are featured.
Jack Edwards is a distinguished mathematician, educator, and researcher whose career has been defined by a deep commitment to making complex mathematical frameworks accessible to the next generation of scholars. With over two decades of experience in both pure mathematics research and university-level instruction, Edwards has established himself as a leading voice in the field of modern geometry. Edwards earned his Ph.D. in Mathematics from Harvard University, specializing in algebraic geometry and arithmetic topology. Prior to his doctoral studies, he completed his Master of Science in Mathematics at the University of Oxford, where he graduated with first-class honours and was awarded the prestigious Senior Mathematical Prize. His foundational academic journey began at Princeton University, where he received his Bachelor’s degree in Mathematics, summa cum laude. Currently serving as a Professor of Mathematics at a top-tier research institution, Edwards has spent years refining the pedagogy of advanced algebra. His research focusing on the intersections of algebraic varieties, sheaf theory, and homological algebra has been widely published in leading international journals, including the Annals of Mathematics and the Journal of the American Mathematical Society. He is also a frequent keynote speaker at global mathematics congresses and a recipient of the Distinguished Teaching Award, an honour that underscores his rare ability to transform abstract, intimidating concepts into elegant and intuitive insights.
| Preface Chapter 1. Overview of Algebraic Geometry Foundations of Algebraic Geometry Emerge from Polynomial Equations Schemes Generalize Varieties through Prime Spectra Birational Geometry Studies Rational Maps between Varieties Homological Mirror Symmetry Conjectures Dualities Noncommutative Geometry Extends Algebraic Methods Abelian Varieties Generalize Elliptic Curves Algebraic Groups Combine Group and Variety Structures Derived Categories and Fourier-Mukai Transforms Vanishing Theorems Control Cohomology Groups Quadratic Forms and Witt Groups Derived Algebraic Geometry and Moduli Chapter 2. Affine and Projective Spaces Affine Space in Algebraic Geometry Projective Space Extends Affine Geometry Naturally Morphisms Connect Different Algebraic Varieties Sheaves Organize Local Data on Algebraic Varieties Intersection Theory Counts Geometric Encounters Classical Algebraic Geometry Solved Ancient Problems Chapter 3. Polynomials and Ideals Polynomial Rings in Algebraic Geometry Hilbert’s Nullstellensatz and its Implications Grobner Bases and Computational Algebra Morphisms between Algebraic Varieties Intersection Theory on Algebraic Varieties Chapter 4. Varieties and Zariski Topology Foundational Concepts of Algebraic Geometry Rational Functions and Function Fields Projective Varieties and Projective Space Singular Points and Smooth Varieties Deformation Theory and Moduli Spaces Arithmetic Geometry and Diophantine Applications Computational Algebraic Geometry Chapter 5. Coordinate Rings and Functions Coordinate Rings as Algebraic Foundations of Geometry Nullstellensatz and Radical Ideals Equations Defining Coordinate Rings Chapter 6. Hilbert Basis Theorem Hilbert Basis Theorem in Algebraic Geometry Geometric Interpretation through Algebraic Varieties Role in Computational Algebraic Geometry Theorem’s Place in Graduate Education Theorem in the Context of Scheme Theory Theorem’s Role in Establishing Noetherian Induction Theorem’s Influence on Ring Theory Development Theorem in the Language of Category Theory |
Chapter 7. Nullstellensatz and Consequences Hilbert’s Nullstellensatz Creates Algebra-Geometry Dictionary Morphisms between Algebraic Sets Correspond to Algebra Homomorphisms Hilbert Polynomials Encode Numerical Invariants of Algebraic Varieties Chow Groups Provide Algebraic Analogue of Homology Theory Hilbert Scheme Parametrizes Subvarieties of Projective Space Derived Algebraic Geometry Enhances Classical Theory with Homotopical Methods Chapter 8. Irreducible Components and Decomposition Concepts of Algebraic Varieties Resolution of Singularities and Component Analysis Relationship to Primary Decomposition in Commutative Algebra Decomposition in Systems Biology and Chemical Reaction Networks Decomposition in Positive Characteristic Chapter 9. Projective Varieties and Homogeneous Ideals Graded Rings and their Algebraic Significance Irrelevant Ideal: A Critical Distinction Hilbert Functions and Polynomials: Numerical Invariants Morphisms of Projective Varieties: Definition and Examples Flag Varieties and Generalized Homogeneous Spaces Canonical Bundles and Adjunction Formulas Toroidal Embeddings and Compactifications Chapter 10. Morphisms between Varieties Concept of Varieties in Algebraic Geometry Graphs of Morphisms and Closed Subvarieties Blowing Up and Birational Modifications Base Change and its Geometric Meaning Hodge Theory and Morphism Induced Maps Coherent Sheaves and Morphism Functoriality Chapter 11. Dimension and Krull Dimension Dimension Theory Begins with Geometric Intuition Prime Ideal Chains Measure Algebraic Depth Constructible Functions Integrate Dimension Data Chapter 12. Regular Functions and Rational Maps Regular Functions Define Algebraic Structure on Varieties Rational Functions Generalize Regular Functions on Varieties Blowups are Birational Maps Replacing Points with Divisors Rational Varieties are Birational to Projective Space Arithmetic Geometry Studies Rational Points on Varieties Chapter 13. Noether Normalization Theorem Noether Normalization Theorem Fundamental Statement Dimension Theory and Krull Dimension Applications to Invariant Theory Generalizations to Schemes and Stacks Connections to Integral Closure and Completion Applications to Birational Geometry Bibliography Index |
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