Algebraic Curves

I. Algebraic Preliminaries.- § 1. Set Theory.- § 2. Integral Domains and Fields.- § 3. Quotient Fields.- § 4. Linear Dependence and Linear Equations.- § 5. Polynomials.- § 6. Factorization in Polynomial Domains.- § 7. Substitution.- § 8. Derivatives.- § 9. Elimination.- §10. Homogeneous Polynomials.- II. Projective Spaces.- § 1. Projective Spaces.- § 2. Linear Subspaces.- § 3. Duality.- § 4. Affine Spaces.- § 5. Projection.- § 6. Linear Transformations.- III. Plane Algebraic Curves.- § 1. Plane Algebraic Curves.- § 2. Singular Points.- § 3. Intersection of Curves.- § 4. Linear Systems of Curves.- § 5. Rational Curves.- § 6. Conies and Cubics.- § 7. Analysis of Singularities.- IV. Formal Power Series.- § 1. Formal Power Series.- § 2. Parametrizations.- § 3. Fractional Power Series.- § 4. Places of a Curve.- § 5. Intersection of Curves.- § 6. Plücker’s Formulas.- § 7. Nöther’s Theorem.- V. Transformations of a Curve.- § 1. Ideals.- § 2. Extensions of a Field.- § 3. Rational Functions ona Curve.- § 4. Birational Correspondence.- § 5. Space Curves.- § 6. Rational Transformations.- § 7. Rational Curves.- § 8. Dual Curves.- § 9. The Ideal of a Curve.- §10. Valuations.- VI. Linear Series.- § 1. Linear Series.- § 2. Complete Series.- § 3. Invariance of Linear Series.- § 4. Rational Transformations Associated with Linear Series.- § 5. The Canonical Series.- § 6. Dimension of a Complete Series.- § 7. Classification of Curves.- § 8. Poles of Rational Functions.- § 9. Geometry on a Non-Singular Cubic.