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.



rational singularities of a certain variety
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](http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.6/share/doc/Macaulay2/Dmodules/html/_has__Rational__Sing_lp__List_rp.html) 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](http://www.personal.psu.edu/kes32/M2.html), 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.


