Let R = C[x,y,u,v] be the coordinate ring of C^4. Let I be the ideal generated by u and v, let J be the ideal generated by x and y. What are the flat deformations of the ideal I^p J^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 C^4.