Exceptional collections for canonical stacks of log del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities

We study derived categories of canonical smooth Deligne-Mumford stacks associated with log del Pezzo surfaces whose singularities are all of type $\frac{1}{3}(1,1)$. While the abstract existence of full exceptional collections for these stacks follows from the Ishii–Ueda special McKay correspondence, our contribution is to make this construction explicit and uniform in the Corti–Heuberger classification. We identify the residual-gerbe object attached to each $\frac{1}{3}(1,1)$-point, derive the resulting length formula, and show how the log del Pezzo cascades turn the problem into the choice of exceptional collections on the six $qG$-rigid base surfaces, followed by Orlov blow-up steps. As an application, we study the sporadic Johnson-Kollár hypersurface $$X_{10}\subset \mathbb{P}(1,2,3,5)\mathrm{.}$$ We identify $X_{10}$ as the anticanonical model of the blow-up of $\mathbb{P}(1,1,3)$ at eight general smooth points, and show that its canonical stack has a full exceptional collection of length 13. We also discuss the corresponding semiorthogonal decomposition of the singular surface via Karmazyn-Kuznetsov-Shinder descent.