Categorical absorption of a non-isolated singularity
Aporva Varshney
TL;DR
The paper addresses a non-isolated compound singularity on a projective threefold and its derived category, introducing a semiorthogonal decomposition that isolates a categorical absorption generated by a compound $P^{\infty}$ object. This absorption mirrors the nodal cases studied by Kuznetsov–Shinder and is shown to arise from a geometric resolution, with the remainder components equivalent to derived categories of smooth varieties; the work also analyzes how the absorption behaves under smoothing through deformation absorption. A detailed analysis on a resolution yields a categorical resolution $\widetilde{D}$ with an explicit exceptional collection, and pushforwards map to the $cP^{\infty}$ object, clarifying the algebra structure and morphisms. Overall, the results extend categorical absorption to a non-isolated setting and demonstrate a robust interaction between resolution, deformation, and absorbing components in the derived category.
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.