The locus of the following curves define affine varieties.
- $xy-z^m=0.$
- $x^m+xy^2+z^2=0.$
- $x^4+y^2+z^2=0.$
- $x^2+y^3+z^2=0.$
- $x^5+y^3+z^2=0.$
In Fulton's "Toric Varieties" book he constructs toric varieties from a fan (a collection of cones satisfying some properties). I am trying to construct the above listed varieties as toric varieties i.e. I am trying to find fans whose associated toric varieties are isomorphic to the affine varieties in the list.
I know how to do the first one. We take the lattice $N= \mathbb{Z}^2$ and let $\sigma$ be the cone generated by $(0,1)$ and $(m,-1).$ Then $\sigma^{\vee}\cap M$ (where $\sigma^{\vee}, M$ are the dual cone and dual lattice) is generated by $(1,0), (1,1), (1,2), \ldots, (1,m).$ Therefore the fan that consists of this cone and all of its faces has the associated toric variety with coordinate ring $\mathbb{C}[X,XY,\cdots, XY^m] \cong \dfrac{ \mathbb{C}[U,V,W] }{(W^m - UV)}$ as required.
Affine varieties must come from a fan of the previous form: A cone and all of its faces. I also know that the resulting varieties from such fans must always be defined by monomial equations i.e. intersections of curves of the form $X_1^{a_1} X_2^{a_2} \ldots X_k^{a_k} = X_1^{b_1} X_2^{b_2} \ldots X_k^{b_k}$ for nonnegative integers $a_i, b_i.$ Varieties 2-5 don't seem to look like this, so I am stuck.
Can someone please help me do the other ones, or refer me to relevant books/papers? Thank you.