Describing a tangent cone. What is that?

Question

Could you please explain what a tangent cone is? For instance, consider the curve on $\mathbb{A}^2$ given by $f(x,y)=x^2-y^3=0$. Linear part is zero cause $\frac{\partial f(0)}{\partial x}=\frac{\partial f(0)}{\partial x}=0$ so the tangent space coincide with all $\mathbb{A}^2$. However, it can be seen that the line $y=0$ also looks like a tangent space (it is some kind of a limit of secants) and the last is called a tangent cone. Ok, but how to describe it explicitly (I mean a description similar to the tangent space one, which is the zero locus of all partial derivatives)? Besides, what is a tangent cone in the case of projective curves (varieties)?

Answer 1

Write $f=\sum\limits_{i=k}^d f_i$, where $f_i$ is homogeneous of degree $i$. The tangent cone at $0$ is the locus $f_k(x,y)=0$.