## Commutative Crepant Resolutions

An important problem in algebraic geometry is the resolution of singularities. Given a singular variety, on wants to construct a proper surjective birational map from a smooth variety to the singular one.

There are many ways to do this but one tries to find a resolution that stays as close as possible to the singularity. An example of such resolutions are crepant resolutions and they are compatible with the notion of a canonical divisor.

In dimension 2 a crepant resolution is unique, but in higher dimensions it is not and it might not even exist. Bridgeland showed however that in dimension 3 although there may be different crepant resolutions, they all have the same homological properties. In other words their derived categories of coherent sheaves are equivalent.

## Noncommutative Crepant Resolutions

In many cases this derived category is equivalent to the derived category of representations of an algebra. Such an algebra is then called a noncommutative crepant resolution (NCCR). As in the commutative case a singularity can have many NCCRs and in general it is a difficult problem to find these for a given singularity.

If we put some restrictions to this the problem becomes more manageble. If \$V\$ is a singular toric 3-dimensional gorenstein variety and A is a toric NCCRs (i.e. an NCCR compatible with the toric structure) I proved that A comes from a dimer model on a torus.

## Dimers

A dimer model can be seen as a quiver drawn on a torus, such that the arrows go round in cycles around pieces of the torus. The algebra \$A\$ is then the path algebra of the quiver subject to a set of relations. For every arrow we identify the two paths in opposite direction on the left and the right of the arrow.

We constructed a program such that for every toric gorenstein 3-singularity spits out all possible dimer models. The toric singularity is determined by the set of ray-vectors of the cone that determines its toric structure as a fan. This data is taken as the input, the output is a set of pictures of the dimer model. The dimer model is pulled back to the universal cover of the torus and gives us a periodic quiver. The fundamental domain is drawn as a dotted square. So for the dimer above you get this picture.

## Fish tilings

Below we give an example of the singularity with as coordinate ring the subring of C[X,Y,Z] spanned by the monomials XaYbZc with ∓a ∓b+c ≥0. The rays are given by the coefficients of the inequalities:

[[1,1,1],[-1,1,1],[1,-1,1],[-1,-1,1]].

There are 4 dimer models. For each we give irs quiver.

For the more artistically minded we also transformed the quiver to a tiling of the plane with fishes. Each fish represents an arrow in the quiver from its tail to its head.