I stumbled upon this isolated singularity of a Calabi-Yau fourfold:
\begin{equation}
x\_1x\_2+x\_3x\_4+x\_5^2=0
\end{equation}
as a hypersurface in $\mathbb{C}^5$.


Clearly, I can resolve this by a simple blow-up. This does not seem to be a crepant resolution however.


So my question is:
Is there some simple criterion which allows me to tell if there exists a crepant resolution ? In case the answer is yes, how would one go about constructing one ?


In the corresponding threefold case there even exists a small resolution. A second question is hence: Are there results on the existence of small resolutions of isolated fourfold singularities ?



Crepant resolution of isolated fourfold singularity
That particular singularity only can have a crepant resolution if it has a small resolution.


Terminal singularities
----------------------


If a variety has terminal singularities (like yours does), the then every further blow-up also has discrepancies $> 0$. (In other words, whether or not the variety has terminal singularities can be read off from a single log resolution). In particular, every discrete valuation has discrepancy $ > 0$. It follows that:


**Fact:** *If a variety has terminal singularities, then it only has a (commutative) crepant resolution if it has a small resolution*


In your example above, the variety clearly has terminal singularities, which can be read off from the blowup you mentioned.


Small resolutions
-----------------


Ok, so now you want to check whether your variety has a small resolution. The first thing you can do is look at [THIS ANSWER](https://mathoverflow.net/questions/31696/best-strategy-for-small-resolutions) by Sándor Kovács. The upshot is there is only hope for a small resolution if your singularity is not factorial.


After staring at your singularity for 4 minutes, I don't see why its not a UFD (hopefully someone, ie Hailong Dao, can tell us easily -- i have a recollection that higher dimensional isolated hypersurface singularities are factorial).


**EDIT:** *Olivier Benoist points out in a comment below that this singularity is factorial due to Samuel's conjecture (proven by Grothendieck).*


However, if you see any given singularity that is not factorial, here's what I would try:


Find a Weil divisor that is not Cartier (the existence of such is equivalent to it being *not* factorial). Blowup the ideal of that Weil divisor. (Maybe do this more than once). The reason I suggest this is because:


**Fact:** *The only way to get a small resolution is to blowup ideals whose vanishing locus has codimension one.*


Here's a quick proof, Blowup an ideal $J$ on a (Noetherian!) variety who vanishing locus $W = V(J)$ has codimension $\geq 2$. The blowup turns that ideal $J$ into a Cartier divisor (it principalizes the ideal). In particular, there is now a divisor lying over $W$. Thus this isn't a small resolution.


Of course, blowing up Weil divisors that are not Cartier is how you get the small resolutions in the $xy-uv = 0$ example.


