Algebraic curves / by Robert J. Walker.
Editor: New York : Dover, 1962Descripción: x, 201 p. ; 21 cmOtra clasificación: 14-01 (14Hxx)CHAPTER I. ALGEBRAIC PRELIMINARIES § 1. Set Theory [3] Sets. Single valued transformations. Equivalence classes. § 2. Integral Domains and Fields [5] Algebraic systems. Integral domains. Fields. Homomorphisms of domains. Exercises. § 3. Quotient Fields [9] § 4. Linear Dependence and Linear Equations [10] Linear dependence. Linear equations. § 5. Polynomials [11] Polynomial domains. The division transformation. Exercise. § 6. Factorization in Polynomial Domains [13] Factorization in domains. Unique factorization of polynomials. Exercises. § 7. Substitution [18] Substitution in polynomials. Zeros of polynomials; the Remainder Theorem. Algebraically closed domains. Exercises. § 8. Derivatives [21] Derivative of a polynomial. Taylor’s Theorem. Exercises. § 9. Elimination [23] The resultant of two polynomials. Application to polynomials in several variables. Exercises. §10. Homogeneous Polynomials [27] Basic properties. Factorization. Resultants. CHAPTER II. PROJECTIVE SPACES § 1. Projective Spaces [32] Projective coordinate systems. Equivalence of coordinate systems. Examples of projective spaces. Exercises. § 2. Linear Subspaces [37] Linear dependence of points. Frame of reference. Linear subspaces. Dimensionality. Relations between subspaces. Exercises. § 3. Duality [41] Hyperplane coordinates. Dual spaces. Dual subspaces. Exercises. § 4. Affine Spaces [44] Affine coordinates. Relation between affine and projective spaces. Subspaces of affine space. Lines in affine space. Exercises. § 5. Projection [47] Projection of points from a subspace. Exercises. § 6. Linear Transformations [48] Collineations. Exercises. CHAPTER III. PLANE ALGEBRAIC CURVES § 1. Plane Algebraic Curves [50] Reducible and irreducible curves. Curves in affine space. Exercises. § 2. Singular Points [52] Intersection of curve and line. Multiple points. Remarks on drawings. Examples of singular points. Exercises. § 3. Intersection of Curves [59] Bezout’s Theorem. Determination of intersections. Exercises. § 4. Linear Systems of Curves [62] Linear systems. Base points. Upper bounds on multiplicities. Exercises. § 5. Rational Curves [66] Sufficient condition for rationality. Exercises. § 6. Conics and Cubics [69] Conics. Cubics. Inflections of a curve. Normal form and flexes of a cubic. Exercises. § 7. Analysis of Singularities [74] 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. CHAPTER IV. FORMAL POWER SERIES § 1. Formal Power Series [87] The domain and the field of formal power series. Substitution in power series. Derivatives. Exercises. § 2. Parametrizations [93] Parametrizations of a curve. Place of a curve. § 3. Fractional Power Series [97] 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 [106] Place with given center. Case of multiple components. Exercises. § 5. Intersection of Curves [108] Order of a polynomial at a place. Intersection of curves. Bezout’s Theorem. Tangent, order, and class of a place. Exercises. § 6. Pliicker’s Formulas [115] Class of a curve. Flexes of a curve. Plücker’s formulas. Exercises. § 7. Nother’s Theorem [120] Nother’s Theorem. Applications. Exercises. CHAPTER V. TRANSFORMATIONS OF A CURVE § 1. Ideals [125] Ideals in a ring. Exercises. § 2. Extensions of a Field [127] Transcendental extensions. Simple algebraic extensions. Algebraic extensions. Exercises. § 3. Rational Functions on a Curve [131] The field of rational functions on a curve. Invariance of the field. Order of a rational function at a place. Exercises. § 4. Birational Correspondence [134] Birational correspondence between curves. Quadratic transformation as birational correspondence. Exercise. § 5. Space Curves [137] Definition of space curve. Places of a space curve. Geometry of space curves. Bezout’s Theorem. Exercises. § 6. Rational Transformations [140] 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 [149] Rational transform of a rational curve. Lüroth’s Theorem. Exercises. § 8. Dual Curves [151] Dual of a plane curve. Plülcker’s formulas. Exercises. § 9. The Ideal of a Curve [155] The ideal of a space curve. Definition of a curve in terms of its ideal. Exercises. §10. Valuations [157] CHAPTER VI. LINEAR SERIES § 1. Linear Series [161] Introduction. Cycles and series. Dimension of a series. Exercises. § 2. Complete Series [165] Virtual cycles. Effective and virtual series. Complete series. Exercises. § 3. Invariance of Linear Series [170] § 4. Rational Transformations Associated with Linear Series . [170] Correspondence between transformations and linear series. Structure of linear series. Normal curves. Complete reduction of singularities. Exercises. § 5. The Canonical Series [176] Jacobian cycles and differentials. Order of canonical series. Genus of a curve. Exercises. § 6. Dimension of a Complete Series [180] Adjoints. Lower bound on dimension. Dimension of canonical series. Special cycles. Theorem of Riemann-Roch. Exercises. § 7. Classification of Curves [186] Composite canonical series. Classification. Canonical forms. Exercises. § 8. Poles of Rational Functions [189] § 9. Geometry on a Non-Singular Cubic [191] Addition of points on a cubic. Tangents. The cross-ratio. Transformations into itself. Exercises.
Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Libros | Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 14 W177-2 (Browse shelf) | Available | A-153 |
"Unabridged and corrected republication of the work first published by Princeton University Press in 1950."
CHAPTER I. ALGEBRAIC PRELIMINARIES --
§ 1. Set Theory [3] --
Sets. Single valued transformations. Equivalence classes. --
§ 2. Integral Domains and Fields [5] --
Algebraic systems. Integral domains. Fields. Homomorphisms of domains. Exercises. --
§ 3. Quotient Fields [9] --
§ 4. Linear Dependence and Linear Equations [10] --
Linear dependence. Linear equations. --
§ 5. Polynomials [11] --
Polynomial domains. The division transformation. Exercise. --
§ 6. Factorization in Polynomial Domains [13] --
Factorization in domains. Unique factorization of polynomials. Exercises. --
§ 7. Substitution [18] --
Substitution in polynomials. Zeros of polynomials; the Remainder Theorem. Algebraically closed domains. Exercises. --
§ 8. Derivatives [21] --
Derivative of a polynomial. Taylor’s Theorem. Exercises. --
§ 9. Elimination [23] --
The resultant of two polynomials. Application to polynomials in several variables. Exercises. --
§10. Homogeneous Polynomials [27] --
Basic properties. Factorization. Resultants. --
CHAPTER II. PROJECTIVE SPACES --
§ 1. Projective Spaces [32] --
Projective coordinate systems. Equivalence of coordinate systems. Examples of projective spaces. Exercises. --
§ 2. Linear Subspaces [37] --
Linear dependence of points. Frame of reference. Linear subspaces. Dimensionality. Relations between subspaces. Exercises. --
§ 3. Duality [41] --
Hyperplane coordinates. Dual spaces. Dual subspaces. Exercises. --
§ 4. Affine Spaces [44] --
Affine coordinates. Relation between affine and projective spaces. Subspaces of affine space. Lines in affine space. Exercises. --
§ 5. Projection [47] --
Projection of points from a subspace. Exercises. --
§ 6. Linear Transformations [48] --
Collineations. Exercises. --
CHAPTER III. PLANE ALGEBRAIC CURVES --
§ 1. Plane Algebraic Curves [50] --
Reducible and irreducible curves. Curves in affine space. Exercises. --
§ 2. Singular Points [52] --
Intersection of curve and line. Multiple points. Remarks on drawings. Examples of singular points. Exercises. --
§ 3. Intersection of Curves [59] --
Bezout’s Theorem. Determination of intersections. Exercises. --
§ 4. Linear Systems of Curves [62] --
Linear systems. Base points. Upper bounds on multiplicities. Exercises. --
§ 5. Rational Curves [66] --
Sufficient condition for rationality. Exercises. --
§ 6. Conics and Cubics [69] --
Conics. Cubics. Inflections of a curve. Normal form and flexes of a cubic. Exercises. --
§ 7. Analysis of Singularities [74] --
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. --
CHAPTER IV. FORMAL POWER SERIES --
§ 1. Formal Power Series [87] --
The domain and the field of formal power series. Substitution in power series. Derivatives. Exercises. --
§ 2. Parametrizations [93] --
Parametrizations of a curve. Place of a curve. --
§ 3. Fractional Power Series [97] --
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 [106] --
Place with given center. Case of multiple components. Exercises. --
§ 5. Intersection of Curves [108] --
Order of a polynomial at a place. Intersection of curves. Bezout’s Theorem. Tangent, order, and class of a place. Exercises. --
§ 6. Pliicker’s Formulas [115] --
Class of a curve. Flexes of a curve. Plücker’s formulas. Exercises. --
§ 7. Nother’s Theorem [120] --
Nother’s Theorem. Applications. Exercises. --
CHAPTER V. TRANSFORMATIONS OF A CURVE --
§ 1. Ideals [125] --
Ideals in a ring. Exercises. --
§ 2. Extensions of a Field [127] --
Transcendental extensions. Simple algebraic extensions. Algebraic extensions. Exercises. --
§ 3. Rational Functions on a Curve [131] --
The field of rational functions on a curve. Invariance of the field. Order of a rational function at a place. Exercises. --
§ 4. Birational Correspondence [134] --
Birational correspondence between curves. Quadratic transformation as birational correspondence. Exercise. --
§ 5. Space Curves [137] --
Definition of space curve. Places of a space curve. Geometry of space curves. Bezout’s Theorem. Exercises. --
§ 6. Rational Transformations [140] --
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 [149] --
Rational transform of a rational curve. Lüroth’s Theorem. Exercises. --
§ 8. Dual Curves [151] --
Dual of a plane curve. Plülcker’s formulas. Exercises. --
§ 9. The Ideal of a Curve [155] --
The ideal of a space curve. Definition of a curve in terms of its ideal. Exercises. --
§10. Valuations [157] --
CHAPTER VI. LINEAR SERIES --
§ 1. Linear Series [161] --
Introduction. Cycles and series. Dimension of a series. Exercises. --
§ 2. Complete Series [165] --
Virtual cycles. Effective and virtual series. Complete series. Exercises. --
§ 3. Invariance of Linear Series [170] --
§ 4. Rational Transformations Associated with Linear Series . [170] --
Correspondence between transformations and linear series. Structure of linear series. Normal curves. Complete reduction of singularities. Exercises. --
§ 5. The Canonical Series [176] --
Jacobian cycles and differentials. Order of canonical series. Genus of a curve. Exercises. --
§ 6. Dimension of a Complete Series [180] --
Adjoints. Lower bound on dimension. Dimension of canonical series. Special cycles. Theorem of Riemann-Roch. Exercises. --
§ 7. Classification of Curves [186] --
Composite canonical series. Classification. Canonical forms. Exercises. --
§ 8. Poles of Rational Functions [189] --
§ 9. Geometry on a Non-Singular Cubic [191] --
Addition of points on a cubic. Tangents. The cross-ratio. Transformations into itself. Exercises. --
MR, 11,387e
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