Let $X$ be an algebraic variety over some algebraically closed field $\Bbbk$ and let us assume $\dim(X)=2$, i.e. $X$ is an algebraic surface.

First, I would like to know the definition of an **ordinary $n$-uple singular point** on $X$, because in the Literature I know, it is only defined with respect to curves. Wolfram Mathworld has a definition, but $X$ does not always admit an embedding into $\mathbb{P}^3$, i.e. it is not necessarily defined by a single equation $f(x,y,z)$, so I am really not sure how this generalizes.

Second, I have been told that such singularities can be resolved by *"blowing up once"* - I would really like to know why that is, i.e. I am looking for a paper or textbook with this statement in it. If it is trivial to proove, of course, that request may be void.