  We consider spline functions over simplicial meshes in $\RR^n$. We assume
that the spline pieces join together with some finite order of smoothness but
the pieces themselves are infinitely smooth. Such splines can have extra orders
of smoothness at a vertex, a property known as \emph{supersmoothness}, which
plays a role in the construction of multivariate splines and in the finite
element method. In this paper we characterize supersmoothness in terms of the
degeneracy of spaces of polynomial splines over the cell of simplices sharing
the vertex, and use it to determine the maximal order of supersmoothness of
various cell configurations.
