Introduction to Algebraic Geometry

Höfundur: Serge Lang (Útgáfa: 0)
Introduction to Algebraic Geometry

Kaup valmöguleikar

Author Serge Lang defines algebraic geometry as the study of systems of algebraic equations in several variables and of the structure that one can give to the solutions of such equations. The study can be carried out in four ways: analytical, topological, algebraico-geometric, and arithmetic. 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.

Nánar um bókina

Útgefandi
Dover Publications
ISBN
9780486839806
Print ISBN
9780486834221
Format
ePub
Útgáfa
0
Höfundar
Serge Lang
Tungumál
English
Útgefið
2019-03-20
Prent takmörkun á líftíma
100

Kaflar

  • Cover
  • Title Page
  • Copyright Page
  • Preface
  • Prerequisites
  • Contents
  • Chapter I General Theory of Places
  • 1. Definition of places
  • 2. Valuations
  • 3. Examples
  • 4. Extension of places
  • 5. Integral closure
  • 6. Places in algebraic extensions
  • Chapter II Algebraic Varieties
  • 1. Notation and preliminaries
  • 2. Decomposition theorem
  • 3. Generic points and specializations
  • 4. The Hilbert Nullstellensatz
  • 5. Decompostion over the algebraic closure of k
  • 6. Product varieties over an algebraically closed field
  • 7. The dimension theorem
  • 8. Homogeneous varieties
  • 9. Elimination theory
  • Chapter III Absolute Theory of Varieties
  • 1. Auxiliary algebraic results
  • 2. Behaviour of an ideal under constant field extensions
  • 3. Absolute varieties, fields of definition, generic points
  • 4. Conjugate varieties
  • 5. The Zariski topology
  • 6. Rationality of a cycle over a field
  • Chapter IV Products, Projections, and Correspondences
  • 1. Products of varieties
  • 2. Projections
  • 3. Rational maps
  • 4. Functions and function fields
  • 5. Correspondences
  • 6. Abstract varieties
  • Chapter V Normal Varieties
  • 1. Integral closure of an affine ring
  • 2. Zariski’s Main Theorem (ZMT)
  • 3. Normalization of an abstract variety
  • 4. Normalization of a projective variety
  • 5. Projective normality
  • 6. Constant field extensions
  • Chapter VI Divisors and Linear Systems
  • 1. Divisors and divisors of functions
  • 2. Existence of functions with given zeros
  • 3. Linear systems
  • 4. The rational map associated with a linear system
  • 5. Divisors rational over a field
  • Chapter VII Differential Forms
  • 1. Derivations
  • 2. Differential forms
  • Chapter VIII Theory of Simple Points
  • 1. Auxiliary results on commutative rings
  • 2. Definition of simple a point
  • 3. Existence of local uniformizing parameters
  • 4. The expansion in power series
  • 5. Dimension theorem for simple components
  • 6. The generic hyperplane section
  • Chapter IX Algebraic Groups
  • 1. Algebraic groups
  • Chapter X Riemann-Rocb Theorem
  • 1. Lemmas on valuations
  • 2. The Riemann-Roch theorem
  • 3. Residues in power series fields
  • 4. The sum of the residues
  • 5. Another proof for the sum of the residues
  • 6. Construction of the Jacobian variety
  • 7. Harnack’s theorem
  • Index