# “Step-by-Step” toric resolution process?

WLOG the fan $\Sigma$ of our toric variety $X_{\Sigma}$ is simplicial. (So $X_{\Sigma}$ has at worst orbifold singularities and all cones $\sigma \in \Sigma$ are simplicial).

The classical toric resolution process shows that there exists a refinement $\Sigma^\prime$ of $\Sigma$ such that $\Sigma^\prime$ is smooth and that the toric morphism $\phi: X_{\Sigma^\prime} \rightarrow X_\Sigma$ is a resolution of singularities.

To prove that such a toric resolution exists, you assign a resolution invariant to each cone $\sigma$ of $\Sigma$ (namely the multiplicity of the cone $\text{mult}(\sigma)$) and produce $\Sigma^\prime$ by a sequence of star subdivision of non-smooth cones. (In geometric terms, these star subdivisions are nothing else then blow-ups) The multiplicity of a cone can be thought of as the size of the orbifoldstrata of the corresponding toric variety, and so a cone $\sigma$ is smooth if and only if $\text{mult}(\sigma)=1$ (so has trivial orbifold group). Moreover, it can easily be shown that the multiplicity $\text{mult}(\sigma)$ of a cone is given by the number of points in $P_\sigma \cap \Lambda$, where $\Lambda$ denotes the lattice and $P_\sigma$ is given by $P_\sigma = \left\{ \sum \limits_{i=1}^{d} \lambda_iu_i : 0 \leq \lambda_i <1 \right\}$

To produce such a sequence of star subdivisions, you do a finite induction over $\text{mult}(\sigma)$. That is, you pick a cone $\sigma \in \Sigma$ with maximal multiplicity and pick an element $v \in P_\sigma \cap \Lambda \setminus \{ 0 \}$. The star subdivision through $v \in \sigma$ replaces all $\tilde\sigma$ such that $v \in \tilde\sigma$ with $\text{Cone}(\tilde\tau,v)$, where $\tilde\tau$ is a face of $\tilde\sigma$ with $v \not\in \tilde\tau$ and it can be shown that $\text{mult}({\text{Cone}(\tau,v)}) < \text{mult}(\sigma)$ for such a face $\tau$ of $\sigma$.

The resolution process does not change the smooth cones but it seems to me that it changes not just the highest dimensional "orbifoldstrata" but also the orbifoldstrata of smaller size (so it also changes the cones of lower multiplicity).

My question is now:

Does there exist a toric resolution process which "just" alters the highest dimensional orbifoldstrata, and leaves the lower dimensional orbifoldstrata untouched?

Or in other words: Is there a "step-by-step" resolution process, which "kills" the orbfioldstrata size by size?

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If not too cumbersome, would you mind giving a precise definition of orbifold groups? – auniket Jun 9 '13 at 17:35
Thank you. I guess "orbifold strata" is more approriate. Shall I clarify what I mean by orbifold strata? – Spinorbundle Jun 9 '13 at 19:56
It would be more clear (at least for me) if you do. – auniket Jun 10 '13 at 13:02