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
16 - Linear Systems
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
A long established philosophy in mathematics is that when you have an interesting set of objects you wish to study you should consider them as the ‘points’ of some ‘space’, and then think in terms of the geometry of that space. The objective of this chapter is to put flesh on this idea, in the case when the interesting objects are projective curves of degree d in PK2. This breaks down into two steps. The first is to extend affine space Kn to projective space PKn by adding points at infinity, generalizing the construction of Chapter 9 given in the case n = 2. And the second step is to show that the curves of degree d in PK2 form a projective space PKD, where D = ½d(d + 3). That enables one to use the machinery of linear algebra to discuss questions about curves. Of particular importance to us will be ‘pencils’ of curves of degree d, corresponding to lines in PKD.
Projective Spaces of Curves
We proceed by analogy with the construction of the projective plane in Section 9.1. We define projective n-space PKn to be the set of lines through the origin O = (0,…,0) in Kn+1. These lines are the points of PKn. Given a non-zero vector X = (x1,…,xn+1) in Kn+1, there is a unique line through X, O determining a unique point in PKn denoted (x1 : … : xn+1).
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- Information
- Elementary Geometry of Algebraic CurvesAn Undergraduate Introduction, pp. 201 - 216Publisher: Cambridge University PressPrint publication year: 1998