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Algebraic curves

Author: Robert John Walker
Publisher: New York ; Berlin : Springer, cop. 1978.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

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  Read more...

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Document Type: Book
All Authors / Contributors: Robert John Walker
ISBN: 0387903615 9780387903613 3540903615 9783540903611
OCLC Number: 490070860
Notes: Reimpr. de l'édition de 1950, publiée par Princeton University Press.
Includes index.
Description: 1 vol. (X-201 p.) ; 24 cm
Contents: 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. Plucker's Formulas.- Class of a curve.- Flexes of a curve.- Plucker's formulas.- Exercises.- 7. Noether's Theorem.- Noether'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.- Luroth's Theorem.- Exercises.- 8. Dual Curves.- Dual of a plane curve.- Plucker'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.
Responsibility: Robert J. Walker.

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