Generation in singularity categories of hypersurfaces of countable representation type

Abstract: The Orlov spectrum and Rouquier dimension are invariants of a triangulated category to measure how big the category is, and they have been studied actively. In this paper, we investigate the singularity category $\mathsf{D_{sg}}(R)$ of a hypersurface $R$ of countable representation type. For a thick subcategory $\mathcal{T}$ of $\mathsf{D_{sg}}(R)$ and a full subcategory $\mathcal{X}$ of $\mathcal{T}$, we calculate the Rouquier dimension of $\mathcal{T}$ with respect to $\mathcal{X}$. Furthermore, we prove that the level in $\mathsf{D_{sg}}(R)$ of the residue field of $R$ with respect to each nonzero object is at most one.