Non-smoothable curve singularities
Jan Stevens
TL;DR
The paper investigates non-smoothable curve singularities by revisiting classic constructions and developing two robust criteria: the Deligne number $e$ and the Buchweitz Dedekind-invariant semicontinuity. It analyzes monomial curves and cones over point sets, employing Gale transforms and Weierstrass semigroup theory to identify non-smoothable instances, including new Gorenstein examples. A large-family argument and explicit computations (often via computer algebra) reveal wide non-smoothable regions, while the Gorenstein and self-associated point-set framework extends non-smoothability to broader families, including $L_{26}^{13}$. The results collectively support the conjecture that generic singularities possess only smooth branches, enriching the understanding of the boundary between smoothable and non-smoothable curve singularities and offering concrete criteria for identifying obstructions in deformation spaces.
Abstract
For curves singularities the dimension of smoothing components in the deformation space is an invariant of the singularity, but in general the deformation space has components of different dimensions. We are interested in the question what the generic singularities are above these components. 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 is specific for curves and uses the semicontinuity of a certain invariant, related to the Dedekind different. This invariant vanishes for Gorenstein, 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 is not smoothable, as the point set cannot be a hyperplane section of a canonical curve, if the genus is at least 11.