NON-SMOOTHABLE CURVE SINGULARITIES

Journal article, 2026

For curves singularities all smoothing components of the deformation space have the same dimension, but there can be components of different dimensions. We are interested in the question of what the generic singularities are that appear in the fibre over a component. To this end we revisit the known examples of non-smoothable singularities and study their deformations. 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 uses the semicontinuity of a certain invariant, related to the Dedekind different. This invariant vanishes for Gorenstein singularities, so in particular for smooth curves. With these methods and computations with computer algebra systems we study monomial curves and cones over point sets in projective space. We also give new explicit examples of non-smoothable singularities. In particular, we find non-smoothable Gorenstein curve singularities. The cone over a general self-associated point set in Pg-2 is not smoothable if g is at least 11, as then the point set can not be a hyperplane section of a canonical curve of genus g.