Crepant resolutions of toric varieties

Question

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

Answer 1

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 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.

Answer 2

I googled a bit, and I found a paper "All toric local complete intersection singularities admit projective crepant resolutions" by Dimitrios I. Dais, Christian Haase, and Günter M. Ziegler. It might be helpful?

It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric local complete intersection singularities. Our strikingly simple proof makes use of Nakajima's classification theorem and of some techniques from toric and discrete geometry.

Answer 3

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.