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  • Produktbild: Algebraic Curves
  • Produktbild: Algebraic Curves

Algebraic Curves

87,99 €

inkl. gesetzl. MwSt., Versandkostenfrei

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

13.12.1978

Verlag

Springer Us

Seitenzahl

201

Maße (L/B/H)

23,5/15,5/1,2 cm

Gewicht

38 g

Auflage

1st ed. 1950. Corr. printing 1978

Sprache

Englisch

ISBN

978-0-387-90361-3

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

13.12.1978

Verlag

Springer Us

Seitenzahl

201

Maße (L/B/H)

23,5/15,5/1,2 cm

Gewicht

38 g

Auflage

1st ed. 1950. Corr. printing 1978

Sprache

Englisch

ISBN

978-0-387-90361-3

Herstelleradresse

Springer-Verlag KG
Sachsenplatz 4-6
1201 Wien
AT

Email: ProductSafety@springernature.com

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  • Produktbild: Algebraic Curves
  • Produktbild: Algebraic Curves
  • 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.