Algebraic Curves

ISBN-13: 9780387903613 / Angielski / Miękka / 1978 / 201 str.

Robert John Walker

Opis

This book was written to furnish a starting point for the study of algebraic geometry. The topics presented and methods of presenting them were chosen with the following ideas in mind; to keep the treat- ment as elementary as possible, to introduce some of the recently devel- oped algebraic methods of handling problems of algebraic geometry, to show how these methods are related to the older analytic and geometric methods, and to apply the general methods to specific geometric prob- lems. These criteria led to a selection of topics from the theory of curves, centering around birational transformations and linear series. Experience in teaching the material showed the need of an intro- duction to the underlying algebra and projective geometry, so this is supplied in the first two chapters. The inclusion of this material makes the book almost entirely self-contained. Methods of presentation, proof of theorems, and problems, have been adapted from various sources. We should mention, in particular, Severi-Laffier, Vorlesungen uber Algebraische Geometrie, van der Waerden, Algebraische Geometrie and Moderne Algebra, and lecture notes of S. Lefschetz and O. Zariski. We also wish to thank Mr. R. L. Beinert and Prof. G. L. Walker for suggestions and assistance with the proof, and particularly Prof. Saunders MacLane for a very careful examination and criticism of an early version of the work. R. J. WALKER Cornell University December 1, 1949 Contents Preface .

Spis treści

I. Algebraic Preliminaries.- § 1. Set Theory.- Sets.- Single valued transformations.- Equivalence classes.- § 2. Integral Domains and Fields.- Algebraic systems.- Integral domains.- Fields.- Homomorphisms of domains.- Exercises.- § 3. Quotient Fields.- § 4. Linear Dependence and Linear Equations.- Linear dependence.- Linear equations.- § 5. Polynomials.- Polynomial domains.- The division transformation.- Exercise.- § 6. Factorization in Polynomial Domains.- Factorization in domains.- Unique factorization of polynomials.- Exercises.- § 7. Substitution.- Substitution in polynomials.- Zeros of polynomials; the Remainder Theorem.- Algebraically closed domains.- Exercises.- § 8. Derivatives.- Derivative of a polynomial.- Taylor’s Theorem.- Exercises.- § 9. Elimination.- The resultant of two polynomials.- Application to polynomials in several variables.- Exercises.- §10. Homogeneous Polynomials.- Basic properties.- Factorization.- Resultants.- II. Projective Spaces.- § 1. Projective Spaces.- Projective coordinate systems.- Equivalence of coordinate systems.- Examples of projective spaces.- Exercises.- § 2. Linear Subspaces.- Linear dependence of points.- Frame of reference.- Linear subspaces.- Dimensionality.- Relations between subspaces.- Exercises.- § 3. Duality.- Hyperplane coordinates.- Dual spaces.- Dual subspaces.- Exercises.- § 4. Affine Spaces.- Affine coordinates.- Relation between affine and projective spaces.- Subspaces of affine space.- Lines in affine space.- Exercises.- § 5. Projection.- Projection of points from a subspace.- Exercises.- § 6. Linear Transformations.- Collineations.- Exercises.- III. Plane Algebraic Curves.- § 1. Plane Algebraic Curves.- Reducible and irreducible curves.- Curves in affine space.- Exercises.- § 2. Singular Points.- Intersection of curve and line.- Multiple points.- Remarks on drawings.- Examples of singular points.- Exercises.- § 3. Intersection of Curves.- Bezout’s Theorem.- Determination of intersections.- Exercises.- § 4. Linear Systems of Curves.- Linear systems.- Base points.- Upper bounds on multiplicities.- Exercises.- § 5. Rational Curves.- Sufficient condition for rationality.- Exercises.- § 6. Conies and Cubics.- Conies.- Cubics.- Inflections of a curve.- Normal form and flexes of a cubic.- Exercises.- § 7. Analysis of Singularities.- Need for analysis of singularities.- Quadratic transformations.- Transformation of a curve.- Transformation of a singularity.- Reduction of singularities.- Neighboring points.- Intersections at neighboring points.- Exercises.- IV. Formal Power Series.- § 1. Formal Power Series.- The domain and the field of formal power series.- Substitution in power series.- Derivatives.- Exercises.- § 2. Parametrizations.- Parametrizations of a curve.- Place of a curve.- § 3. Fractional Power Series.- The field K(x)* of fractional power series.- Algebraic closure of K(x)*.- Discussion and example.- Extensions of the basic theorem.- Exercises.- § 4. Places of a Curve.- Place with given center.- Case of multiple components.- Exercises.- § 5. Intersection of Curves.- Order of a polynomial at a place.- Intersection of curves.- Bezout’s Theorem.- Tangent, order, and class of a place.- Exercises.- § 6. Plücker’s Formulas.- Class of a curve.- Flexes of a curve.- Plücker’s formulas.- Exercises.- § 7. Nöther’s Theorem.- Nöther’s Theorem.- Applications.- Exercises.- V. Transformations of a Curve.- § 1. Ideals.- Ideals in a ring.- Exercises.- § 2. Extensions of a Field.- Transcendental extensions.- Simple algebraic extensions.- Algebraic extensions.- Exercises.- § 3. Rational Functions on a Curve.- The field of rational functions on a curve.- Invariance of the field.- Order of a rational function at a place.- Exercises.- § 4. Birational Correspondence.- Birational correspondence between curves.- Quadratic transformation as birational correspondence.- Exercise.- § 5. Space Curves.- Definition of space curve.- Places of a space curve.- Geometry of space curves.- Bezout’s Theorem.- Exercises.- § 6. Rational Transformations.- Rational transformation of a curve.- Rational transformation of a place.- Example.- Projection as a rational transformation.- Algebraic transformation of a curve.- Exercises.- § 7. Rational Curves.- Rational transform of a rational curve.- Lüroth’s Theorem.- Exercises.- § 8. Dual Curves.- Dual of a plane curve.- Plücker’s formulas.- Exercises.- § 9. The Ideal of a Curve.- The ideal of a space curve.- Definition of a curve in terms of its ideal.- Exercises.- §10. Valuations.- VI. Linear Series.- § 1. Linear Series.- Cycles and series.- Dimension of a series.- Exercises.- § 2. Complete Series.- Virtual cycles.- Effective and virtual series.- Complete series.- Exercises.- § 3. Invariance of Linear Series.- § 4. Rational Transformations Associated with Linear Series.- Correspondence between transformations and linear series.- Structure of linear series.- Normal curves.- Complete reduction of singularities.- Exercises.- § 5. The Canonical Series.- Jacobian cycles and differentials.- Order of canonical series.- Genus of a curve.- Exercises.- § 6. Dimension of a Complete Series.- Adjoints.- Lower bound on dimension.- Dimension of canonical series.- Special cycles.- Theorem of Riemann-Roch.- Exercises.- § 7. Classification of Curves.- Composite canonical series.- Classification.- Canonical forms.- Exercises.- § 8. Poles of Rational Functions.- § 9. Geometry on a Non-Singular Cubic.- Addition of points on a cubic.- Tangents.- The cross-ratio.- Transformations into itself.- Exercises.