Small resolution of a non-isolated singularity?

Question

Consider the the Hypersurface singularity given by the equation $$xyz+st=0 \subset \mathbb{C}^5.$$ How would you describe a (nice!=symmetric) small-resolution of this singularity?

Answer 1

There are six small crepant resolutions of this singularity, it being the (pullback via the Weyl group of the) versal deformation space of the $\mathbb{Z}_3$ surface singularity. The six small resolutions are in one-to-one correspondence with the Weyl group, namely $S_3$, and they correspond to orderings of $\{x,y,z\}$.

To see how this works, if you blow-up the ideal $(s,x)$ you'll get a space with two affine opens; one is $\mathbb{A}^4$, the other is in some co-ordinates $yz=st$ inside $\mathbb{A}^5$. For the second chart you have two options; blowup $(s,y)$ or $(s,z)$. I write the first option as $xyz$ (since we blew up the ideal $(s,x)$ first, then $(s,y)$, then $z$ is left), and I write the second option as $xzy$.

Clearly we could have started instead by blowing up $(s,y)$; this leads to orderings $yxz$ and $yzx$. You get orderings $zxy$ and $zyx$ in a similar way.

All the small crepant resolutions are related by flops (in fact they are all derived equivalent and nice things happen, but this is irrelevant here), and in this context flops just correspond to permuting the orderings. For example, xyz flops to yxz (permute the first two variables) and also to xzy (permute the second two variables).

In my mind there is no reason as to why any of the orderings $xyz$, $xzy$, $yxz$, $yzx$, $zxy$, $zyx$ are distinguished or indeed "most symmetric". From an abstract geometric perspective, I would say none are better than any of the others. Indeed, exactly the same thing happens in one dimension lower for the singularity $xy=st\subset \mathbb{A}^4$. For it, there are two small crepant resolutions, corresponding to orderings $xy$ and $yx$, and neither are "distinguished" or "prefered".