5 - Singularities

Summary

INTRODUCTION

We have met curves that aren't everywhere smooth. For example in ℝ², the curve y² = x³ has a cusp at the origin, and in a neighborhood of the origin the alpha curve y² = x²(x + 1) is ×-shaped. Each of these points is a singularity of the curve. The term “singular” connotes exceptional or rare. Within any particular complex affine or projective curve, singular points are indeed rare because there are only finitely many of them among the infinitely many points of the curve. A curve having no singularities is called nonsingular.

Singular points are rare in yet another way: most algebraic curves have no singularities at all! That is to say, if we randomly choose coefficients of p(x, y) then C(p) in ℂ² or ℙ²(ℂ)is nonsingular. “Random” has the same meaning as in Chapter 1: a general polynomial p(x, y) of degree n has finitely many coefficients, and since p and any nonzero multiple of it define the same curve, in randomly picking each of these finitely many coefficients, we may choose our dartboard to be the interval (−1, 1) ⊃ ℝ for a polynomial in ℝ[x, y], or from the unit disk about 0 ∈ ℂ for a polynomial in ℂ[x, y].

In spite of their rarity, singular points can be found in curves defined by very simple polynomials, and understanding these special points can reveal quite a bit about the nature of algebraic curves in general. Important concepts in mathematics usually have both geometric and algebraic counterparts, and that's true of singular points.