It is well known that the dimension of multivariate spline spaces depends not just on the toplogy of the underlying triangulation but also on its geometry, i.e., the precise location of the vertices.
A standard way of utilizing spline spaces for the solution of interpolation problems is to minimize a suitable functional subject to the interpolation conditions. Now consider the location of the vertices variable, and in particular consider a sitatuion where the vertices move into a position where the dimesnion of the spline space increases. In the limit there is an additional degree of freedom, and onw would expect the solution of the interpolation/minimization problem to change discontinuously. It turns out that sometimes this happens and sometimes it does not. We would like to understnad this situation much better, and we would also like to know how to handle it numerically.
You may wish to look at a more technical description and explanation of the term commutation problem.