Toric Desingularization Algorithms There are certainly many algorithms to desingularize toric varieties (e.g https://arxiv.org/pdf/math/0411340.pdf). I would imagine in analogy with desingularizing surfaces these all involve blowing up successively in some way, which translates to iteratively subdividing cones in the fan that defines the toric variety.
Can anyone provide references to find which algorithms might be most helpful if I wanted to compute some properties of the desingularization, and ideally would like to create as few new cones as possible?
 A: The answer of your question may change depending on what you mean with "as few new cones as possible". For instance, you may be interested in minimize the amount of maximal dimensional cones, or minimize the number of one dimensional cones, or try to minimize the number of all cones of all dimensions.
Let me point out what is usually the most natural way to resolve toric varieties: Let suppose that your toric variety is given by $X(\sigma)={\rm Spec}(\mathbb{K}[\sigma^\vee \cap M])$, where $\sigma\subset M$ is a rational polyhedral cone, then what you would like to do is to find a Hilbert basis for the monoid $\sigma^\vee\cap M$ with certain algorithm. The rays between the origin and the elements of the Hilbert basis will give you the codimension-one structure of the resolution of singularities. However, I don't think that there is always an "obvious" or "natural" way to choose the higher codimension structure.
For example, consider the cone generated by $(1,0,0),(0,1,0),(0,0,1)$ and $(-1,1,1)$. Cleary you can resolve this singularity by adding the ray $(0,1,1)$
and then introducing the obvious four $2$-dimensional faces. But also, you can just resolve this toric singularity by adding a single $2$-dimensional face.
If you tell me what exactly you want to minimize, I may be able to give a more complete answer.
