Algebraic Curves

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 .

I. Algebraic Preliminaries.- § 1. Set Theory.- § 2. Integral Domains and Fields.- § 3. Quotient Fields.- § 4. Linear Dependence and Linear Equations.- § 5. Polynomials.- § 6. Factorization in Polynomial Domains.- § 7. Substitution.- § 8. Derivatives.- § 9. Elimination.- §10. Homogeneous Polynomials.- II. Projective Spaces.- § 1. Projective Spaces.- § 2. Linear Subspaces.- § 3. Duality.- § 4. Affine Spaces.- § 5. Projection.- § 6. Linear Transformations.- III. Plane Algebraic Curves.- § 1. Plane Algebraic Curves.- § 2. Singular Points.- § 3. Intersection of Curves.- § 4. Linear Systems of Curves.- § 5. Rational Curves.- § 6. Conies and Cubics.- § 7. Analysis of Singularities.- IV. Formal Power Series.- § 1. Formal Power Series.- § 2. Parametrizations.- § 3. Fractional Power Series.- § 4. Places of a Curve.- § 5. Intersection of Curves.- § 6. Plücker’s Formulas.- § 7. Nöther’s Theorem.- V. Transformations of a Curve.- § 1. Ideals.- § 2. Extensions of a Field.- § 3. Rational Functions ona Curve.- § 4. Birational Correspondence.- § 5. Space Curves.- § 6. Rational Transformations.- § 7. Rational Curves.- § 8. Dual Curves.- § 9. The Ideal of a Curve.- §10. Valuations.- VI. Linear Series.- § 1. Linear Series.- § 2. Complete Series.- § 3. Invariance of Linear Series.- § 4. Rational Transformations Associated with Linear Series.- § 5. The Canonical Series.- § 6. Dimension of a Complete Series.- § 7. Classification of Curves.- § 8. Poles of Rational Functions.- § 9. Geometry on a Non-Singular Cubic.