# $Z/pZ$ - acyclic affine toric varieties

Let me ask you a question about $Z/pZ$ - acyclic affine toric varieties (for some $p$) i.e. toric varieties $X$ such that homologies $H_{j}(X,Z/pZ) = 0$ for $j > 0$ and $H_{0}(X,Z/pZ) = Z/pZ$. I am interested how big such class of $Z/pZ$ - acyclic affine varieties is? For example, it is known that the affine spaces are the examples of such $Z/pZ$ - acyclic varieties. On the other hand, are $(C^{*})^r \times (C^{+})^{n-r}$ $Z/pZ$ - acyclic varieties? So, again:

Question: how big class of $Z/pZ$ - acyclic affine toric varieties is?

If the associated cone has empty interior, equivalently, if there are no torus-invariant points, then there will be a subgroup of the torus which the stabilizer of every point is contained in. Taking the quotient by this subgroup will give a map from your algebraic variety to the quotient torus with connected fibers. One can pull back a nontrivial cohomology class in $H^1$ along this map, and it will remain nontrivial.