I. Algebraic Varieties in a Projective Space.- I. Fundamental Concepts.- ??? 1. Plane Algebraic Curves.- 1. Rational Curves.- 2. Connections with the Theory of Fields.- 3. Birational Isomorphism of Curves.- Exercises.- ???2. Closed Subsets of Affine Spaces.- 1. Definition of Closed Subset.- 2. Regular Functions on a Closed Set.- 3. Regular Mappings.- Exercises.- ??? 3. Rational Functions.- 1. Irreducible Sets.- 2. Rational Functions.- 3. Rational Mappings.- Exercises.- ??? 4. Quasiprojective Varieties.- 1. Closed Subsets of ...
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I. Algebraic Varieties in a Projective Space.- I. Fundamental Concepts.- ??? 1. Plane Algebraic Curves.- 1. Rational Curves.- 2. Connections with the Theory of Fields.- 3. Birational Isomorphism of Curves.- Exercises.- ???2. Closed Subsets of Affine Spaces.- 1. Definition of Closed Subset.- 2. Regular Functions on a Closed Set.- 3. Regular Mappings.- Exercises.- ??? 3. Rational Functions.- 1. Irreducible Sets.- 2. Rational Functions.- 3. Rational Mappings.- Exercises.- ??? 4. Quasiprojective Varieties.- 1. Closed Subsets of a Projective Space.- 2. Regular Functions.- 3. Rational Functions.- 4. Examples of Regular Mappings.- Exercises.- ??? 5. Products and Mappings of Quasiprojective Varieties.- 1. Products.- 2. Closure of the Image of a Projective Variety.- 3. Finite Mappings.- 4. Normalization Theorem.- Exercises.- ??? 6. Dimension.- 1. Definition of Dimension.- 2. Dimension of an Intersection with a Hypersurface.- 3. A Theorem on the Dimension of Fibres.- 4. Lines on Surfaces.- 5. The Chow Coordinates of a Projective Variety.- Exercises.- II. Local Properties.- ???1. Simple and Singular Points.- 1. The Local Ring of a Point.- 2. The Tangent Space.- 3. Invariance of the Tangent Space.- 4. Singular Points.- 5. The Tangent Cone.- Exercises.- ???2. Expansion in Power Series.- 1. Local Parameters at a Point.- 2. Expansion in Power Series.- 3. Varieties over the Field of Real and the Field of Complex Numbers 88 Exercises.- ??? 3. Properties of Simple Points.- 1. Subvarieties of Codimension 1.- 2. Smooth Subvarieties.- 3. Factorization in the Local Ring of a Simple Point.- Exercises.- ??? 4. The Structure of Birational Isomorphisms.- 1. The ?-Process in a Projective Space.- 2. The Local ?-Process.- 3. Behaviour of Subvarieties under a ?-Process.- 4. Exceptional Subvarieties.- 5. Isomorphism and Birational Isomorphism.- Exercises.- ???5. Normal Varieties.- 1. Normality.- 2. Normalization of Affine Varieties.- 3. Ramification.- 4. Normalization of Curves.- 5. Projective Embeddings of Smooth Varieties.- Exercises.- III. Divisors and Differential Forms.- ??? 1. Divisors.- 1. Divisor of a Function.- 2. Locally Principal Divisors.- 3. How to Shift the Support of a Divisor Away from Points.- 4. Divisors and Rational Mappings.- 5. The Space Associated with a Divisor.- Exercises.- ??? 2. Divisors on Curves.- 1. The Degree of a Divisor on a Curve.- 2. Bezout's Theorem on Curves.- 3. Cubic Curves.- 4. The Dimension of a Divisor.- Exercises.- ???3. Algebraic Groups.- 1. Addition of Points on a Plane Cubic Curve.- 2. Algebraic Groups.- 3. Factor Groups. Chevalley's Theorem.- 4. Abelian Varieties.- 5. Picard Varieties.- Exercises.- ???4. Differential Forms.- 1. One-Dimensional Regular Differential Forms.- 2. Algebraic Description of the Module of Differentials.- 3. Differential Forms of Higher Degrees.- 4. Rational Differential Forms.- Exercises.- ??? 5. Examples and Applications of Differential Forms.- 1. Behaviour under Mappings.- 2. Invariant Differential Forms on a Group.- 3. The Canonical Class.- 4. Hypersurfaces.- 5. Hyperelliptic Curves.- 6. The Riemann-Roch Theorem for Curves.- 7. Projective Immersions of Surfaces.- Exercises.- IV. Intersection Indices.- ???1. Definition and Basic Properties.- 1. Definition of an Intersection Index.- 2. Additivity of the Intersection Index.- 3. Invariance under Equivalence.- 4. End of the Proof of Invariance.- 5. General Definition of the Intersection Index.- Exercises.- ???2. Applications and Generalizations of Intersection Indices.- 1. Bezout's Theorem in a Projective Space and Products of Projective Spaces.- 2. Varieties over the Field of Real Numbers.- 3. The Genus of a Smooth Curve on a Surface.- 4. The Ring of Classes of Cycles.- Exercises.- ??? 3. Birational Isomorphisms of Surfaces.- 1. ?-Processes of Surfaces.- 2. Some Intersection Indices.- 3. Elimination of Points of Indeterminacy.
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1977 revised printing, 439 pp., hardcover, NEW! ! . -If you are reading this, this item is actually (physically) in our stock and ready for shipment once ordered. We are not bookjackers. Buyer is responsible for any additional duties, taxes, or fees required by recipient's country.
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New. Trade paperback (US). Glued binding. 440 p. Contains: Unspecified. Springer Study Edition. In Stock. 100% Money Back Guarantee. Brand New, Perfect Condition, allow 4-14 business days for standard shipping. To Alaska, Hawaii, U.S. protectorate, P.O. box, and APO/FPO addresses allow 4-28 business days for Standard shipping. No expedited shipping. All orders placed with expedited shipping will be cancelled. Over 3, 000, 000 happy customers.
