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Algebraic Curves over a Finite Field /
This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correctin...
| Autores principales: | , , |
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| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Princeton, New Jersey :
Princeton University Press,
2008.
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| Colección: | Book collections on Project MUSE.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Title; Copyright; Dedication; Contents; Preface; PART 1. GENERAL THEORY OF CURVES; Chapter 1. Fundamental ideas; 1.1 Basic definitions; 1.2 Polynomials; 1.3 Affine plane curves; 1.4 Projective plane curves; 1.5 The Hessian curve; 1.6 Projective varieties in higher-dimensional spaces; 1.7 Exercises; 1.8 Notes; Chapter 2. Elimination theory; 2.1 Elimination of one unknown; 2.2 The discriminant; 2.3 Elimination in a system in two unknowns; 2.4 Exercises; 2.5 Notes; Chapter 3. Singular points and intersections; 3.1 The intersection number of two curves; 3.2 Bezout's Theorem.
- 3.3 Rational and birational transformations3.4 Quadratic transformations; 3.5 Resolution of singularities; 3.6 Exercises; 3.7 Notes; Chapter 4. Branches and parametrisation; 4.1 Formal power series; 4.2 Branch representations; 4.3 Branches of plane algebraic curves; 4.4 Local quadratic transformations; 4.5 Noether's Theorem; 4.6 Analytic branches; 4.7 Exercises; 4.8 Notes; Chapter 5. The function field of a curve; 5.1 Generic points; 5.2 Rational transformations; 5.3 Places; 5.4 Zeros and poles; 5.5 Separability and inseparability; 5.6 Frobenius rational transformations.
- 5.7 Derivations and differentials5.8 The genus of a curve; 5.9 Residues of differential forms; 5.10 Higher derivatives in positive characteristic; 5.11 The dual and bidual of a curve; 5.12 Exercises; 5.13 Notes; Chapter 6. Linear series and the Riemann-Roch Theorem; 6.1 Divisors and linear series; 6.2 Linear systems of curves; 6.3 Special and non-special linear series; 6.4 Reformulation of the Riemann-Roch Theorem; 6.5 Some consequences of the Riemann-Roch Theorem; 6.6 The Weierstrass Gap Theorem; 6.7 The structure of the divisor class group; 6.8 Exercises; 6.9 Notes.
- Chapter 7. Algebraic curves in higher-dimensional spaces7.1 Basic definitions and properties; 7.2 Rational transformations; 7.3 Hurwitz's Theorem; 7.4 Linear series composed of an involution; 7.5 The canonical curve; 7.6 Osculating hyperplanes and ramification divisors; 7.7 Non-classical curves and linear systems of lines; 7.8 Non-classical curves and linear systems of conics; 7.9 Dual curves of space curves; 7.10 Complete linear series of small order; 7.11 Examples of curves; 7.12 The Linear General Position Principle; 7.13 Castelnuovo's Bound; 7.14 A generalisation of Clifford's Theorem.
- 7.15 The Uniform Position Principle7.16 Valuation rings; 7.17 Curves as algebraic varieties of dimension one; 7.18 Exercises; 7.19 Notes; PART 2. CURVES OVER A FINITE FIELD; Chapter 8. Rational points and places over a finite field; 8.1 Plane curves defined over a finite field; 8.2 Fq-rational branches of a curve; 8.3 Fq-rational places, divisors and linear series; 8.4 Space curves over Fq; 8.5 The Stöhr-Voloch Theorem; 8.6 Frobenius classicality with respect to lines; 8.7 Frobenius classicality with respect to conics; 8.8 The dual of a Frobenius non-classical curve; 8.9 Exercises; 8.10 Notes.