MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 3 down vote accepted

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:

$\mathbf{1)}$ 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);

$\mathbf{2})$ 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 $-n$.

share|cite|improve this answer
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. – rita Aug 10 '11 at 8:20
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 – Francesco Polizzi Aug 10 '11 at 9:10
@Francesco: in this Eprint, 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. – rita Aug 10 '11 at 15:43
@Rita: Ok. Thank you very much for the link. – Francesco Polizzi Aug 10 '11 at 19:05
What are $x$ and $y$? Does $\hat{\mathcal{O}}_P$ denote the completion of the local ring at $P$? – Jesko Hüttenhain Aug 12 '11 at 21:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.