User:Coffeebrake60/sandbox

Plane curve singularities


Historically, singularities were first noticed in the study of real algebraic curves. The double point at (0,0) of the curve $$y^2 = x^2 - x^3\ $$ and the cusp point at (0,0) of the curve $$y^2 = x^3\ $$ are qualitatively different, as is seen just by sketching. In more precise terms, if we take out the point (0,0) in both curves, in the first case we obtain two connected components and in the second case three connected components. Isaac Newton carried out a detailed study of all cubic curves, the general family to which these examples belong.

Consider now the curves defined over the field of complex numbers by the same equations. They are surfaces of $$R^4 $$ and it is not very easy to visualize them. Isaac Newton initiated the local study of complex analytical plane curves. He developed a method to parametrize them, what is now called the Puiseux expansion of the curve. The Puiseux expansions of the curves given by the equations

$$y^2 - x^3=0\ $$ and $$(\ y^2 - x^3)^2- ...\ $$

are respectively $$\ y = x^{3/2}\ $$ and $$\ y = x^{3/2}+x^{7/4}.$$

This means that the curves referred above admit the parametrizations

$$f(t)=(t^2,t^3)\ $$ and   $$\ g(t)=(t^4,t^6+t^7)\ $$