  We discuss the effect of perturbations on the ground rings of $c=1$ string
theory at the various compactification radii defining the $A_N$ points of the
moduli space. We argue that perturbations by plus-type moduli define ground
varieties which are equivalent to the unperturbed ones under redefinitions of
the coordinates and hence cannot smoothen the singularity. Perturbations by the
minus-type moduli, on the other hand, lead to semi-universal deformations of
the singular varieties that can smoothen the singularity under certain
conditions. To first order, the cosmological perturbation by itself can remove
the singularity only at the self-dual ($A_1$) point.}

High Energy Physics - Theory