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Let $X$ be a variety in $\mathbb{C}^{10}$ defined by the ideal $I=\left<xz'-x'z, y'(u+z)-y(u'+z'), t'(u-z)-t(u'-z')+xy'-x'y\right>$ of $\mathbb{C}[x,y,z,u,t,x',y',z',u',t']$. Note that $I+\left<u-z, u'-z'\right>$ gives a determinantal variety $D$ defined over the matrix

$ \left( \begin{array}{ccc} x & y & z \\ x' & y' & z' \end{array} \right).$

It is well-known that $D$ has rational singularities. Is there any machinery theorem to deduce the property of having rational singularities for $X$ from that of $D$?

Any reference is greatly appreciated. Thanks in advance.

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If $u-z, u'-z'$ happen to be a regular sequence on $\mathbb{C}[x,..., t']/I$, then this follows by Elkik's result on deformation of rational singularities (at least then $X$ is rational near $D$). Otherwise I think you are out of luck in terms of general tools. –  Karl Schwede Sep 22 '13 at 22:35
    
However, you might be able to check this using a computer... (let me know if I could help with this). –  Karl Schwede Sep 22 '13 at 22:35
    
I checked this with a computer, it appears to be rational. –  Karl Schwede Sep 23 '13 at 2:41
    
Dear Karl, thanks so much for your answers. They are not a regular sequence because dim(D)=dim(X)-1. Anyway, it's very helpful that you know it has rational singularities. Could you please show me how to verify it? Thanks again. –  NN guest Sep 23 '13 at 9:47
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1 Answer

up vote 5 down vote accepted

Ok, $X$ is a complete intersection (dimension 7, cut out by three equations in $\mathbb{C}^{10}$).

You can then use the hasRationalSing function described HERE in the D-modules package of Macaulay2.

You can also verify it by the following method. Use the F-singularities PosChar.m2 package which can be downloaded HERE, choose a characteristic (I choose 3), define the same singularity, and then run the isFRegularQGor(R/I, 1) command. This proves that the equation is F-regular and hence F-rational and thus pseudo-rational in characteristic 3. It should follow from some general result (that I'll have to track down a precise reference for that later, it should be contained in a dissertation from the University of Michigan last year) that this implies that R/I has rational singularities in characteristic 0.

In this case, I checked both commands and they both agreed that it had rational singularities.

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This is cool! Whose thesis was that? –  Hailong Dao Sep 23 '13 at 15:09
    
I think it was the thesis of Zhixian Zhu, in particular it should follow from a variant of Corollary 4.2 of arxiv.org/pdf/1308.5445v1.pdf I thought she was graduating last year, but maybe not (she's still listed as a student). –  Karl Schwede Sep 23 '13 at 16:32
    
Many thanks, Karl! This helps a ton. –  NN guest Sep 24 '13 at 9:32
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