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 Bézout's Theorem -- 3.3 Rational and birational transformations -- 3.4 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 differentials -- 5.8 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 spaces -- 7.1 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 Principle -- 7.16 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 -- Chapter 9. Zeta functions and curves with many rational points -- 9.1 The zeta function of a curve over a finite field -- 9.2 The Hasse-Weil Theorem -- 9.3 Refinements of the Hasse-Weil Theorem -- 9.4 Asymptotic bounds -- 9.5 Other estimates -- 9.6 Counting points on a plane curve -- 9.7 Further applications of the zeta function -- 9.8 The Fundamental Equation -- 9.9 Elliptic curves over Fq -- 9.10 Classification of non-singular cubics over Fq -- 9.11 Exercises -- 9.12 Notes -- PART 3. FURTHER DEVELOPMENTS -- Chapter 10. Maximal and optimal curves -- 10.1 Background on maximal curves.

10.2 The Frobenius linear series of a maximal curve -- 10.3 Embedding in a Hermitian variety -- 10.4 Maximal curves lying on a quadric surface -- 10.5 Maximal curves with high genus -- 10.6 Castelnuovo's number -- 10.7 Plane maximal curves -- 10.8 Maximal curves of Hurwitz type -- 10.9 Non-isomorphic maximal curves -- 10.10 Optimal curves -- 10.11 Exercises -- 10.12 Notes -- Chapter 11. Automorphisms of an algebraic curve -- 11.1 The action of K-automorphisms on places -- 11.2 Linear series and automorphisms -- 11.3 Automorphism groups of plane curves -- 11.4 A bound on the order of a K-automorphism -- 11.5 Automorphism groups and their fixed fields -- 11.6 The stabiliser of a place -- 11.7 Finiteness of the K-automorphism group -- 11.8 Tame automorphism groups -- 11.9 Non-tame automorphism groups -- 11.10 K-automorphism groups of particular curves -- 11.11 Fixed places of automorphisms -- 11.12 Large automorphism groups of function fields -- 11.13 K-automorphism groups fixing a place -- 11.14 Large p-subgroups fixing a place -- 11.15 Notes -- Chapter 12. Some families of algebraic curves -- 12.1 Plane curves given by separated polynomials -- 12.2 Curves with Suzuki automorphism group -- 12.3 Curves with unitary automorphism group -- 12.4 Curves with Ree automorphism group -- 12.5 A curve attaining the Serre Bound -- 12.6 Notes -- Chapter 13. Applications: codes and arcs -- 13.1 Algebraic-geometry codes -- 13.2 Maximum distance separable codes -- 13.3 Arcs and ovals -- 13.4 Segre's generalisation of Menelaus' Theorem -- 13.5 The connection between arcs and curves -- 13.6 Arcs in ovals in planes of even order -- 13.7 Arcs in ovals in planes of odd order -- 13.8 The second largest complete arc -- 13.9 The third largest complete arc -- 13.10 Exercises -- 13.11 Notes -- Appendix A. Background on field theory and group theory -- A.1 Field theory.

A.2 Galois theory -- A.3 Norms and traces -- A.4 Finite fields -- A.5 Group theory -- A.6 Notes -- Appendix B. Notation -- Bibliography -- Index.