For a plane curve $C$ defined by $F(x,y)$ containing $0$, we say $0$ is a node if one of the following conditions satisfied:
- The tangent cone (i.e. the variety defined by the lowest degree part of $F$) is defined by non-degenerate quadratic form
- The completion of local ring $\hat{\mathcal O}_{C,0}$ is isomorphic to $k[[x,y]]/(Q)$ where $Q$ non-degenerate quadratic.
I want to know why they are equivalent?
For simplicity, let us assume that we are working over an algebraically closed field. Then, any non-degenerate quadratic form can be assumed to be $xy$ after a change of variables. In $k[[x,y]]$, a power series of the form $xy+$ higher order terms can be seen to be product of two power series of the form $x+$ higher order and $y+$ higher order. So, these two can be taken as the generators of the power series ring and so we are done. Converse is just using the identification $k[x,y]/(x,y)^3=k[[x,y]]/(x,y)^3$.