TITLE: Crepant resolutions of toric varieties
QUESTION [9 upvotes]: Given a toric variety, is it easy to see if a crepant resolution exists?  If so, how can it be explicitly constructed?

REPLY [5 votes]: The first requirement is that the toric variety is $\mathbb Q$-Gorenstein, otherwise discrepancies and crepancy are not defined. Combinatorially, this means that for every cone of the fan $\Sigma$ the generators of the exceptional rays lie in a hyperplane. 
A resolution of such cone is crepant iff all new rays added lie on the aforementioned hyperplane for the cone of the original fan that they sit in. Nonsingularity is a statement that the finer fan is simplicial and unimodular.
While simpliciality is rather easy to satisfy, unimodularity is unlikely in dimension four or higher. It can be managed in dimension up to three by Pick's theorem.
One possible solution to this issue is to consider stacky resolutions which are smooth toric DM stacks that map birationally to the original toric variety. Then all one needs is a $\mathbb Q$-Gorenstein condition.