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. $x^5+y^3+z^2=0.$

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. $x^5+y^3+z^2=0.$

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.

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. $x^5+y^3+z^2=0.$