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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.

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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, 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 $-n$.

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    $\begingroup$ I think {\em ordinary $n$-tuple point} is also sometimes used for hypersurface singularities that are the cone over a smooth plane curve. For instance, the elliptic singularity $x^3+y^3+z^3=0$ is also called an ordinary triple point. $\endgroup$
    – rita
    Commented Aug 10, 2011 at 8:20
  • $\begingroup$ Dear Rita, this is also reasonable. Do you have any reference? Maybe Miyaoka meant that the ones mentioned in my answer are the only ordinary $n$-tuple points among quotient singularities $\endgroup$
    – Francesco Polizzi
    Commented Aug 10, 2011 at 9:10
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    $\begingroup$ @Francesco: in this Eprint front.math.ucdavis.edu/0010.5163, ordinary triple point is used in the sense I mentioned. The only case that satisfies both definitions is a singularity of type $A_1$, since the cone over $C_n$ is not a Gorenstein singularity for $n>2$. I think both definitions exist in the literature. $\endgroup$
    – rita
    Commented Aug 10, 2011 at 15:43
  • $\begingroup$ @Rita: Ok. Thank you very much for the link. $\endgroup$
    – Francesco Polizzi
    Commented Aug 10, 2011 at 19:05
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    $\begingroup$ @rita the link in your comment is broken, here is a replacement: arxiv.org/abs/math/0010163 $\endgroup$
    – David Roberts ♦
    Commented Mar 29, 2022 at 7:03

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