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.