Let $\Delta$ be a fan, and $X(\Delta)$ the toric variety.
The formula in section 7 here asserts that $|X(\mathbb{F}_q)| = \sum_{k= 0}^n (q - 1)^k d_{n - k}$, where $d_j$ is the numberof $j$ dimensional faces of $\Delta$, provided that $X$ is smooth.
On the other hand, looking at page 94. in Fulton, the given reference, it is not so clear that this smoothness condition is necessary. It seems that the argument that decomposes $X$ into the torus orbits, each of which contributes $(q - 1)^{dimension}$ points should go through. However, the discussion in Fulton is informal, and I'm concerned that I'm missing something.
Question: What requirements on $X$ are necessary for that point counting formula to be true? Is there a reference where this theorem is stated explicitly?
Comments : We counted points for some $2$-dimensional singular toric varieties and got agreement. (E.g. $p^2$ points for the quadric cone.)
The counting argument only uses the orbit-cone correspondence, and if I'm not mistaken it holds without smoothness assumption.