If $ X $ is a toric variety and one has a closed sub-variety $ Y \subseteq X $, is the blow-up $ \operatorname{Bl}\_{Y}(X) $ a toric variety as well? I suspect not, but wanted to check with other users to see if someone had a counterexample or a proof. If my suspicions are correct, when is the blow-up a toric variety? 



When is the blow-up of a closed sub-variety of a toric variety a toric variety?
Usually the answer is no. Here are a couple of ways of proving $Bl\_Y(X)$ is not toric. I'll work over $\mathbb{C}$.


Example 1. Suppose $Bl\_Y(X)$ is a smooth and projective toric variety. Then the cycle class map
$$
CH^\*(Bl\_Y(X)) \rightarrow H^\*(Bl\_Y(X),\mathbb{Z})
$$
is an isomorphism. In particular, smooth toric varieties only have cohomology in even degrees. Now if $Y$ is a smooth curve with genus$(Y)\ge 1$ and $X$ is a threefold, then $Bl\_Y(Z)$ will satisfy
$$
H^3(Bl\_Y(Z),\mathbb{Z}) \cong H^1(Y,\mathbb{Z}) \ne 0.
$$


Example 2. The effective cone of curves (the Mori cone) of a toric variety is always finitely generated. On the other hand, if you blow up enough points in general position in $\mathbb{P}^2$ (a toric variety), the resulting surface will have countably many (-1)-curves, which are always extremal in the effective cone. Thus the effective cone will be at least countably generated, and $Bl\_Y(X)$ is not a toric variety.


Now if we assume that the blow up map
$$
\pi\colon Bl\_Y(X) \rightarrow X
$$
is a map of toric varieties - i.e. the action of $(\mathbb{C}^\*)^n$ commutes with $\pi$ - then the ideal of the exceptional locus will push forward to a $(\mathbb{C}^\*)^n$-fixed ideal in $\mathcal{O}\_X$. Locally in the affine charts associated to the toric variety, any such ideal will be generated by "monomials." Visa versa, the blow up of a $(\mathbb{C}^\*)^n$-fixed ideal is a toric variety.


In general though, I don't know of any characterization of subvarieties $Y\subset X$ such that the blow up of $Y$ is abstractly a toric variety.


