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

Mathematics > Algebraic Geometry [Submitted on 16 Jun 2026] Title:Exceptional collections for canonical stacks of log del Pezzo surfaces with $\frac13(1,1)$ singularities View PDF HTML (experimental)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 $\pi:\mathcal X\to X$ and compare it with 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 $D^b(\operatorname{coh}\mathcal X)$ admits a full exceptional collection. The proof combines rationality of log del Pezzo surfaces, Orlov's blow-up formula, and the special McKay correspondence of Ishii--Ueda. We then specialize to a general degree $10$ hypersurface $X_{10}\subset \mathbb P(1,2,3,5)$. The Corti--Heuberger cascade identifies its minimal resolution as $\widetilde{X}_{10}\cong \operatorname{Bl}8\mathbb F_3$, and therefore the canonical stack $\mathcal X_{10}$ has a full exceptional collection of length $13$. We also discuss the singular coarse category through the approach of Karmazyn--Kuznetsov--Shinder. Submission history From: Alex Junior Gomez Saltachin [view email][v1] Tue, 16 Jun 2026 17:58:07 UTC (13 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.