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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?

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Just out of curiosity, how do you know that one exists? –  Hailong Dao Sep 24 '13 at 12:57
    
I think this is a toric singularity (a cone given by a fan which is itself a cone over a cylinder with triangular base) and we can minimally resolve it be subdiving this fan. But the result wont be symmetric. I am looking for a non-toric approach which would give me a symmetric picture (symmetric w.r.t to x,y,z), if possible at all. –  Mohammad F. Tehrani Sep 24 '13 at 17:03
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Not every toric singularity admits a small resolution. Every toric singularity admits a small modification to a toric variety having only quotient singularities. But, as you say, that modification may not be unique. –  Jason Starr Sep 24 '13 at 17:17
    
You are right, but I think this one does and is given by a triangulation of the shape I described. –  Mohammad F. Tehrani Sep 24 '13 at 18:38

1 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".

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Hi Michael, welcome to MO! –  Hailong Dao Sep 25 '13 at 12:18
    
Thanks. All the mentioned resolutions above are toric and correspond to different triangulations of the fan described in my comment. Thus, my question is why these are the only small resolutions. I would appreciate if you expend the first sentence of your answer into more details (where I think the answer to my question is hidden). –  Mohammad F. Tehrani Sep 25 '13 at 14:38

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