Torsion Differentials and Deformation

Let -scheme be a Schlessinger deformation of a curve defined over a field . In §§1 and 2, the dimension of the parameter space , the relative differentials of over , and the fibres with singularity were studied, in case when is locally complete-intersection. In §3 we show that if -scheme is a specialization of a smooth -scheme, then the punctured spectrum <!-- MATH $\operatorname{Spex} ({O_{{X_{0,x}}}})$ --> has to be connected for every point <!-- MATH $x \in {X_0}$ --> such that <!-- MATH $\dim {O_{{X_{0,x}}}} \geqslant 2$ --> . In turn we construct a rigid singularity on a surface. In the last section a few conjectures amplifying those of P. Deligne are made.