Seminar in Algebraic Geometry and Number Theory

Abstract: Not all singularities are smoothable, as there exist rigid ones. It is conjectured that curve singularities are never rigid. Curve singularities stand out because they always can be parametrised. Moreover, the dimension of smoothing components in the deformation space is an invariant of the singularity, computable via the parametrisation. In general the deformation space has components of different dimensions. We are interested in the question what the generic singularities are above these components.
 
There are two general methods available to show that a curve is not smoothable. In the first method one exhibits a family of singularities of a certain type and then uses a dimension count to prove that the family cannot lie in the closure of the space of smooth curves. The other method is specific for curves and uses the semicontinuity of a certain invariant, related to the Dedekind different. This invariant vanishes for locally Gorenstein, so in particular for smooth curves. Direct computation, mostly for quasi-homogeneous singularities of relatively small codimension and degree, leads to general conjectures.