Book contents
- Frontmatter
- Contents
- List of Illustrations
- List of Tables
- Preface
- 1 Real Algebraic Curves
- 2 General Ground Fields
- 3 Polynomial Algebra
- 4 Affine Equivalence
- 5 Affine Conics
- 6 Singularities of Affine Curves
- 7 Tangents to Affine Curves
- 8 Rational Affine Curves
- 9 Projective Algebraic Curves
- 10 Singularities of Projective Curves
- 11 Projective Equivalence
- 12 Projective Tangents
- 13 Flexes
- 14 Intersections of Projective Curves
- 15 Projective Cubics
- 16 Linear Systems
- 17 The Group Structure on a Cubic
- 18 Rational Projective Curves
- Index
18 - Rational Projective Curves
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- List of Illustrations
- List of Tables
- Preface
- 1 Real Algebraic Curves
- 2 General Ground Fields
- 3 Polynomial Algebra
- 4 Affine Equivalence
- 5 Affine Conics
- 6 Singularities of Affine Curves
- 7 Tangents to Affine Curves
- 8 Rational Affine Curves
- 9 Projective Algebraic Curves
- 10 Singularities of Projective Curves
- 11 Projective Equivalence
- 12 Projective Tangents
- 13 Flexes
- 14 Intersections of Projective Curves
- 15 Projective Cubics
- 16 Linear Systems
- 17 The Group Structure on a Cubic
- 18 Rational Projective Curves
- Index
Summary
In this final chapter, we return to the question of the ‘rationality’ of curves, which we touched upon briefly (in the affine case) in Chapter 8. It is more natural to study the concept in the projective case, since the behaviour of the curve ‘at infinity’ plays a role. Ideally, one seeks a necessary and sufficient condition for a curve to be rational. We would need to develop rather more algebra to expose the elegant answer to this question provided by curve theory. Our compromise is to show that a useful class of curves (those of ‘deficiency’ zero) are rational, and that a substantial class of curves (the non-singular ones of degree ≥ 3) fail to be rational. These results require most of the machinery we have developed in this text, and provide good illustrations of the underlying techniques.
The Projective Concept
The idea of a ‘rational’ curve can be extended from the affine plane to the projective plane in a very natural way. An irreducible projective curve F of degree d in PK2 is rational when there exist forms X(s,t), Y(s,t), Z(s,t) of the same degree d for which the following conditions hold.
(i) For all but finitely many values of the ratio (s : t), at least one of the scalars X(s,t), Y(s,t), Z(s,t) is non-zero, and we have F(X(s,t),Y(s,t),Z(s,t)) = 0.
[…]
- Type
- Chapter
- Information
- Elementary Geometry of Algebraic CurvesAn Undergraduate Introduction, pp. 234 - 246Publisher: Cambridge University PressPrint publication year: 1998