Rigid ideals by deforming quadratic letterplace ideals

We compute the deformation space of quadratic letterplace ideals $L(2,P)$ of finite posets $P$ when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal $J(2,P)$ defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases $J(2,P)$ is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.