On deformations of maps and curve singularities

We study several deformation functors associated to the normalization of a reduced curve singularity . The main new results are explicit formulas, in terms of classical invariants of ( X , 0), for the cotangent cohomology groups T i , i = 0,1,2, of these functors. Thus we obtain precise statements about smoothness and dimension of the corresponding local moduli spaces. We apply the results to obtain explicit formulas, respectively, estimates for the -codimension of a parametrized curve singularity, where denotes the Mather–Wall group of left-right equivalence.