The Affine Tangent Cone I'm failing to see how exactly is the tangent cone at a singular point on a curve picking out all the different tangent lines through this singular point (say the origin in $\mathbb{A}^2$)?
Could someone explain this, or at least redirect me to a source I could read about? I tried to look online, but at most places they are just taking this as a known fact!
Thanks!

For anyone interested I will write a reference for the 'answer' to this question: It turns out that this is a problem in Hartshorne, namely Problem 5.14 (b) of Chapter 1. I will write the statement here, and maybe at some point post a solution as well.
If $f=f_r+f_{r+1}+\cdots \in k[[ x, y]]$, and if the leading form $f_r$ factors as $f_r=g_sh_t$, where $g_s,h_t$ are homogeneous of degrees $s$ and $t$ respectively, and have no common linear factors, then there are formal power series 
$$g=g_s+g_{s+1}+\cdots$$
$$h=h_t+h_{t+1}+\cdots$$ in $k[[x,y]]$ such that $f=gh.$
