I am trying to understand singular points on a complex projective -algebraic curve. I remember that singular points on a affine algebraic curve are determined by taking the partial derivatives and finding where they equal to 0.
But then I found this definition of singular point on a complex projective-algebraic curve that says that the multiplicity $v_p(C)$ of a curve C at p is the order of the lowest non-vanishing term in the taylor expansion of f at p. And thus a point is singular when $v_p(C) >1$. This process just confuses me and I wondered if I can use the process for affine curves.
I'm just confused on why these are different, for example for the curve $x^2-y^3$, why can I not set this equal to 0 , find the partial derivatives and tell me this has a singular point at $(0,0)$.
Lastly the textbook I'm using mentions that examples for singular points are at double points and simple cusps. And explain that the equation for a double point is $xy =0$ which I'm not sure how this would help if I saw a curve.