Given a toric variety, is it easy to see if a crepant resolution exists? If so, how can it be explicitly constructed?



Crepant resolutions of toric varieties
Whether or not a resolution is crepant only depends on the hypersurfaces in the exceptional locus -- to speak casually, it depends on which hypersurfaces you add. In the toric case, resolution corresponds to subdividing the fan, and the new hypersurfaces correspond to the new rays you add. In particular, if I can resolve without adding any new rays, that resolution will certainly be crepant.


What is going to happen is that there will be some combinatorial rule picking out a finite collection of rays. This will be fairly explicit and simple. You will then have to determine whether there is a subdivision of the original fan to a smooth fan, adding in only these permitted rays. That is a difficult combinatorial question, although it is finite in principal. 


Also, whether or not a ray is permitted will depend solely on the cone of the original fan that it is in. (Geometrically, the crepancy of a divisor can be computed locally on the variety being resolved.) So it is enough to, for a given polyhedral cone, give a rule for which rays in that cone are permitted.


Now I'm going to disappoint you by not remembering that rule. The only case I remember is where sigma is a simplicial cone, with generators v\_1, v\_2, ..., v\_n, and there is some monomial w in the dual lattice such that <w, v\_1>= <w, v\_2> = ... = <w, v\_n> = 1. Then a ray, with minimal lattice vector v, corresponds to a crepant divisor if and only if <w, v>=1. This condition is actually fairly natural; it is the case of a quotient singularity of the form C^n/G where G is a finite abelian subgroup of SL\_n. 


Everything I know about this, I learned during the 2006 Michigan [working seminar](http://www.math.lsa.umich.edu/seminars/alggeo/topicsAG/Winter06.shtml#124) on the McKay correspondence. You might find the references there to be useful.


UPDATE: I thought a bit harder, and I can state the above rule slightly more generally: The cone sigma doesn't have to be simplicial. It is enough that there is some monomial in the dual lattice which pairs with the generators this way. If the cone is not simplicial, then this gives more linear equations for w than we have variables, but they might happen to be solvable anyway. This is precisely the situation that the singularity is Gorenstein.


