Let $n$ be a positive integer and let $A$ be the subring of ${\mathbb C}[x,y]$ generated 
by $x,xy,...,xy^n$. Let $S=Spec(A)$. This is an affine surface, which is clearly singular if
$n\neq 1$. Is this some sort of familiar surface? For example, is it normal? Does it have rational singularities? Can one construct an explicit resolution of singularities for it?



An affine singular surface
Let $\mathbb{F}\_n$ be the Hirzebruch surface of index $n$. You have two open subsets on it isomorphic to $\mathbb{A}^1\times \mathbb{P}^1$, with a glueing map given by $$(t,[u:v])\mapsto (1/t,[ut^n:v])$$
The section $u=0$ has self-intersection $-n$ and the section $v=0$ has self-intersection $n$. We remove the section $v=0$ and obtain an affine surface if $n>0$ (because the section is ample). The morphism to your surface given by
 $$(t,[u:v])\mapsto (\frac{u}{v},\frac{u}{v}t,\dots,\frac{u}{v}t^n)$$
on the first chart corresponds to 
$$(t,[u:v])\mapsto (\frac{u}{v}t^n,\frac{u}{v}t^{n-1},\dots,\frac{u}{v}t,\frac{u}{v})$$
on the other chart. Hence the resolution of your singularity is just the complement in $\mathbb{F}\_n$ of one section of self-intersection $n$, and the exceptional divisor is the section of self-intersection $-n$.


The surface obtained is called a Gizatullin surface, because of the very nice theorem of Gizatullin that asserts that the isomorphism class of the complement of an ample section in a Hirzebruch surface only depends on the square of the section (and not of the section or the Hirzebruch surface).


