
Basic Algebraic Geometry 1
Varieties in Projective Space
Igor R. Shafarevich(Author)
Springer (Publisher)
2nd Edition
Published on 8. August 1994
Book
Paperback/Softback
XX, 304 pages
978-3-540-54812-6 (ISBN)
Article exhausted; check for reprint
Description
The first edition of this book came out just as the apparatus of algebraic geometry was reaching a stage that permitted a lucid and concise account of the foundations of the subject. The author was no longer forced into the painful choice between sacrificing rigour of exposition or overloading the clear geometrical picture with cumbersome algebraic apparatus. The 15 years that have elapsed since the first edition have seen the appear ance of many beautiful books treating various branches of algebraic geometry. However, as far as I know, no other author has been attracted to the aim which this book set itself: to give an overall view of the many varied aspects of algebraic geometry, without going too far afield into the different theories. There is thus scope for a second edition. In preparing this, I have included some additional material, rather varied in nature, and have made some small cuts, but the general character of the book remains unchanged.
Reviews / Votes
From the reviews:
"To my best knowledge, only this book manages to describe so many advanced constructions while still being accessible for researchers outside the field of algebraic geometry. This book is indeed a tremendous achievement." (Newsletter on Computational and Applied Mathematics)
More details
Edition
2nd, rev. and exp. ed. 1994
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer Berlin
Target group
College/higher education
Graduate
Edition type
Revised edition
Product notice
Paperback (trade)
Illustrations
1
Dimensions
Height: 23.5 cm
Width: 15.5 cm
Thickness: 17 mm
Weight
1010 gr
ISBN-13
978-3-540-54812-6 (9783540548126)
DOI
10.1007/978-3-642-57908-0
Schweitzer Classification
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Basic Algebraic Geometry 1
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Content
I. Basic Notions.- 1. Algebraic Curves in the Plane.- 1.1. Plane Curves.- 1.2. Rational Curves.- 1.3. Relation with Field Theory.- 1.4. Rational Maps.- 1.5. Singular and Nonsingular Points.- 1.6. The Projective Plane.- Exercises to §1.- 2. Closed Subsets of Affine Space.- 2.1. Definition of Closed Subsets.- 2.2. Regular Functions on a Closed Subset.- 2.3. Regular Maps.- Exercises to §2.- 3. Rational Functions.- 3.1. Irreducible Algebraic Subsets.- 3.2. Rational Functions.- 3.3. Rational Maps.- Exercises to §3.- 4. Quasiprojective Varieties.- 4.1. Closed Subsets of Projective Space.- 4.2. Regular Functions.- 4.3. Rational Functions.- 4.4. Examples of Regular Maps.- Exercises to §4.- 5. Products and Maps of Quasiprojective Varieties.- 5.1. Products.- 5.2. The Image of a Projective Variety is Closed.- 5.3. Finite Maps.- 5.4. Noether Normalisation.- Exercises to §5.- 6. Dimension.- 6.1. Definition of Dimension.- 6.2. Dimension of Intersection with a Hypersurface.- 6.3. The Theorem on the Dimension of Fibres.- 6.4. Lines on Surfaces.- Exercises to §6.- II. Local Properties.- 1. Singular and Nonsingular Points.- 1.1. The Local Ring of a Point.- 1.2. The Tangent Space.- 1.3. Intrinsic Nature of the Tangent Space.- 1.4. Singular Points.- 1.5. The Tangent Cone.- Exercises to §1.- 2. Power Series Expansions.- 2.1. Local Parameters at a Point.- 2.2. Power Series Expansions.- 2.3. Varieties over the Reals and the Complexes.- Exercises to §2.- 3. Properties of Nonsingular Points.- 3.1. Codimension 1 Subvarieties.- 3.2. Nonsingular Subvarieties.- Exercises to §3.- 4. The Structure of Birational Maps.- 4.1. Blowup in Projective Space.- 4.2. Local Blowup.- 4.3. Behaviour of a Subvariety under a Blowup.- 4.4. Exceptional Subvarieties.- 4.5. Isomorphism and Birational Equivalence.- Exercises to §4.- 5. Normal Varieties.- 5.1. Normal Varieties.- 5.2. Normalisation of an Affine Variety.- 5.3. Normalisation of a Curve.- 5.4. Projective Embedding of Nonsingular Varieties.- Exercises to §5.- 6. Singularities of a Map.- 6.1. Irreducibility.- 6.2. Nonsingularity.- 6.3. Ramification.- 6.4. Examples.- Exercises to §6.- III. Divisors and Differential Forms.- 1. Divisors.- 1.1. The Divisor of a Function.- 1.2. Locally Principal Divisors.- 1.3. Moving the Support of a Divisor away from a Point ....- 1.4. Divisors and Rational Maps.- 1.5. The Linear System of a Divisor.- 1.6. Pencil of Conics over ?1.- Exercises to §1.- 2. Divisors on Curves.- 2.1. The Degree of a Divisor on a Curve.- 2.2. Bézout's Theorem on a Curve.- 2.3. The Dimension of a Divisor.- Exercises to §2.- 3. The Plane Cubic.- 3.1. The Class Group.- 3.2. The Group Law.- 3.3. Maps.- 3.4. Applications.- 3.5. Algebraically Nonclosed Field.- Exercises to §3.- 4. Algebraic Groups.- 4.1. Algebraic Groups.- 4.2. Quotient Groups and Chevalley's Theorem.- 4.3. Abelian Varieties.- 4.4. The Picard Variety.- Exercises to §4.- 5. Differential Forms.- 5.1. Regular Differential 1-forms.- 5.2. Algebraic Definition of the Module of Differentials.- 5.3. Differential p-forms.- 5.4. Rational Differential Forms.- Exercises to §5.- 6. Examples and Applications of Differential Forms.- 6.1. Behaviour Under Maps.- 6.2. Invariant Differential Forms on a Group.- 6.3. The Canonical Class.- 6.4. Hypersurfaces.- 6.5. Hyperelliptic Curves.- 6.6. The Riemann-Roch Theorem for Curves.- 6.7. Projective Embedding of a Surface.- Exercises to §6.- IV. Intersection Numbers.- 1. Definition and Basic Properties.- 1.1. Definition of Intersection Number.- 1.2. Additivity.- 1.3. Invariance Under Linear Equivalence.- 1.4. The General Definition of Intersection Number.- Exercises to §1.- 2. Applications of Intersection Numbers.- 2.1. Bézout's Theorem in Projective and Multiprojective Space.- 2.2. Varieties over the Reals.- 2.3. The Genus of a Nonsingular Curve on a Surface.- 2.4. The Riemann-Roch Inequality on a Surface.- 2.5. The Nonsingular Cubic Surface.- 2.6. The Ring of Cycle Classes.- Exercises to §2.- 3. Birational Maps of Surfaces.- 3.1. Blowups of Surfaces.- 3.2. Some Intersection Numbers.- 3.3. Resolution of Indeterminacy.- 3.4. Factorisation as a Chain of Blowups.- 3.5. Remarks and Examples.- Exercises to §3.- 4. Singularities.- 4.1. Singular Points of a Curve.- 4.2. Surface Singularities.- 4.3. Du Val Singularities.- 4.4. Degeneration of Curves.- Exercises to §4.- Algebraic Appendx.- References.