# Rational surface singularities as Toric varieties

I originally asked basically this question on MSE here. From an appendix in Steinberg's "Conjugacy classes in algebraic groups" I found a list of the rational surface singularities. Equations of types $A_n, D_n, E_6, E_7$ and $E_8$ were given:

1. $xy-z^m=0.$
2. $x^m+xy^2+z^2=0.$
3. $x^4+y^2+z^2=0.$
4. $x^2+y^3+z^2=0.$
5. $x^5+y^3+z^2=0.$

What I would like to do is construct (varieties isomorphic to) these as Toric varieties of a fan (as Fulton does it) or prove that it is not possible. As you can see from the link to my previous question, I know how to create the one of type $A_n$ but not the others. I suspect it is not possible but I don't know how to prove that. Can anyone help me resolve this?

Thank you very much.

Sorry, my list is different to Steinberg's, I can't remember where my incorrect list is from then. Steinberg's list:

1. $xy+z^{n+1}=0.$
2. $x^{n+1}+xy^2+z^2=0.$
3. $x^4+y^3+z^2=0.$
4. $x^3y+y^3+z^2=0.$
5. $x^5+y^3+z^2=0.$
• Are you sure the equations you've written for the $E_6$ and $E_7$ singularities are correct? – user5117 Oct 23 '13 at 11:33
• @ArtiePrendergast-Smith Thank you for pointing that out, I've corrected it. – Katie Dobbs Oct 23 '13 at 11:53

It's not. These are constructible as ${\mathbb A}^2/\Gamma$ for $\Gamma$ a finite subgroup of $SL(2)$, and you can recover $\Gamma$ as $\pi_1$ of a punctured neighborhood of the singularity. If you compute that group for a toric surface singularity, it's abelian, whereas for your $D,E$ examples it's not.
• I am not very familiar with this material. Do you mind giving me some extra details? Specifically I would like to know what the $\Gamma$ are, why $\Gamma$ is the $\pi_1$ of a punctured nbhd of the singular point and why for punctured nbhds of toric surface singularities $\pi$ must be abelian (I realised this list is everything you said unfortunately :( ). I realise I'm asking a lot, I'd be grateful if you could point me to some references for those facts if you don't want to type up the details. Thank you for your help! EDIT: I can probably find a list of the $\Gamma$ on my own actually. – Katie Dobbs Oct 23 '13 at 12:04
• If you puncture ${\mathbb C}^2$, you get something that retracts to $S^3$, a simply-connected space on which $\Gamma$ acts without fixed points. As for the toric side, when the edges coming out of a vertex are a $\mathbb Q$-basis, and so generate a subgroup of ${\mathbb Z}^n$ with finite cokernel $\Gamma^*$, a neighborhood of the point is isomorphic to ${\mathbb A}^n / \Gamma$. I think this is in [Fulton, Intro to TVs] under "rational smoothness". – Allen Knutson Oct 23 '13 at 15:55