Let $R = \mathbf{C}[x,y,u,v]$ be the coordinate ring of $\mathbf{C}^4$. Let $I$ be the ideal generated by $u$ and $v$, let $J$ be the ideal generated by $u$ and $y$. What are the flat deformations of the ideal $I^pJ^q$?

This is the ideal of the union of two non-transversely intersecting $2$-planes, with some nilpotent thickness on each plane. I would like to learn something about its smooth deformations as a subvariety of $\mathbf{C}^4$.