Torsion differentials and deformation

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