Exceptional collections for canonical stacks of log del Pezzo surfaces with $\frac13(1,1)$ singularities

Abstract

We study derived categories associated with log del Pezzo surfaces whose singularities are of type $\frac13(1,1)$. The Ishii--Ueda special McKay correspondence, together with the rationality of the minimal resolution, provides the structural mechanism for constructing full exceptional collections on the canonical stack. We identify the residual-gerbe contribution, obtain an explicit length formula, and use the Corti--Heuberger cascades to give a uniform construction procedure once a full exceptional collection on the resolution of the rigid base is fixed. For the sporadic Johnson--Koll\'ar hypersurface $X_{10}\subset \mathbf P(1,2,3,5)$, we use the Reid--Suzuki realization as the anticanonical model of the blow-up of $\mathbf P(1,1,3)$ at eight general smooth points to construct an explicit full exceptional collection of length $13$. We compute its Euler form and obtain a semiorthogonal decomposition of the singular surface with a $K(3,1)$-component and ten exceptional objects via Karmazyn--Kuznetsov--Shinder descent. Finally, we determine the Brauer group throughout the six Corti--Heuberger cascades: it is trivial for $1\leq k\leq 5$ and isomorphic to $\mathbf Z/3$ in the six-point cascade.