Determinantal Singularities

Abstract

We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena such as for instance non-isolated singularities which are finitely determined, or smoothings with low connectivity; already the union of the coordinate axes in (C 3 , 0) is determinantal, but not a complete intersection. We start with the algebraic background and then continue by discussing the subtle interplay of unfoldings and deformations in this setting, including a survey of the case of determinantal hypersurfaces, Cohen-Macaulay codimension 2 and Gorenstein codimension 3 singularities, and determinantal rational surface singularities. We conclude with a discussion of essential smoothings and provide an appendix listing known classifications of simple determinantal singularities.

On the topology of determinantal links

“…In this case the determinantal deformations coincide with the usual deformations of the singularity. We refer to [FKZ21] for the details on determinantal singularities and their deformations and further references for the above statements.…”

“…, t k . See, for instance, [FKZ21] for the underlying notion of GL-equivalence and the existence and construction of such miniversal unfoldings. The induced deformation…”