# rational singularities of a certain variety

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

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