Categorical absorption of a non-isolated singularity

Abstract

We study an example of a projective threefold with a non-isolated singularity and its derived category. The singular locus can be locally described as a line of surface nodes compounded with a threefold node at the origin. We construct a semiorthogonal decomposition where one component absorbs the singularity in the sense of Kuznetsov–Shinder, and the other components are equivalent to the derived categories of smooth projective varieties. The absorbing category is seen to be closely related to the absorbing category constructed for nodal varieties by Kuznetsov–Shinder, reflecting the geometry of the singularity. We further show that the semiorthogonal decomposition is induced by one on a geometric resolution and briefly consider the properties of the absorbing category under smoothing.