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](http://mathworld.wolfram.com/OrdinaryDoublePoint.html), 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.



Ordinary n-uple Points and Resolution of Singularities on a Surface
A germ $(X, p)$ of isolated surface singularity is called an *ordinary n-tuple point* if
$$\hat{\mathcal{O}}\_p=\mathbb{C}[[ x^n, x^{n-1}y, \ldots, xy^{n-1}, y^n]],$$
see for instance Miyaoka's paper [The maximal number of quotient singularities on surfaces with given numerical invariants](https://doi.org/10.1007/BF01456083 "Math. Ann. 268, 159–171 (1984), EuDML:182912. zbMATH review at https://zbmath.org/0521.14013"), Section 1.


Equivalently, this can be seen as:


1. the singularity type (at its vertex) of the cone $C\_n$ over the rational normal curve of degree $n$ in $\mathbb{P}^n$. In particular the embedding dimension is $n+1$, so it cannot be realized as a isolated singularity in $\mathbb{P}^3$ unless $n=2$ (ordinary double point);
2. the cyclic quotient singularity $\frac{1}{n}(1,1)$, i.e. the quotient of $\mathbb{C}^2$ by the action of the group $\mathbf{Z}/n \mathbf{Z}$ given by $$\xi \cdot (x,\ y) \to (\xi x, \ \xi y),$$ where $\xi$ is a primitive $n$-th root of unity. In particular it is a *rational* singularity.


The fact that such singularities can be resolved by "blowing up once" is a standard computation. Or, if you prefer, just note that the blow up of the cone $C\_n$ at its vertex is the Hirzebruch surface $\mathbf{F}\_n$, which is smooth. This also shows that the minimal resolution of an ordinary $n$-tuple point is given by a smooth rational curve with self-intersection $-n$.


