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 \(\frac{1}{3}(1,1)\). For such a surface \(X\), we consider the canonical smooth Deligne--Mumford stack \(π\colon \mathcal{X} \to X\) and also discuss the singular coarse surface \(X\). Our main result proves that, if \(X\) is a complex log del Pezzo surface whose singularities are all of type \(\frac{1}{3}(1,1)\), then \(\mathrm{D}^{\mathrm{b}}(\operatorname{coh} \mathcal{X})\) admits a full exceptional collection. We also give an explicit description of the local exceptional objects supported on the residual gerbes of the stacky points. As an application, we study a general degree \(10\) hypersurface \(X_{10} \subset \mathbb{P}(1,2,3,5)\), one of the sporadic Johnson--Kollár examples. We show that its canonical stack \(\mathcal{X}_{10}\) has a full exceptional collection of length \(13\), and we discuss the corresponding singular coarse category.