Orbifold Degenerations of Hirzebruch Surfaces

[2608.27652] Orbifold Degenerations of Hirzebruch Surfaces Skip to main content Search Submit Donate Log in Mathematics > Algebraic Geometry arXiv:2608.27652 (math) [Submitted on 27 Aug 2026] Title: Orbifold Degenerations of Hirzebruch Surfaces Authors: Juan Pablo Zúñiga View PDF HTML (experimental) Abstract: We study orbifold degenerations of Hirzebruch surfaces. Our main theorem shows that every such degeneration $X$ arises as a partial smoothing of a toric surface. We then give a combinatorial description of the singularities that arise when $-K_X$ is not nef. This complements previous joint work with G. Urzúa, which treated the case in which $-K_X$ is ample. For Hirzebruch surfaces $\mathbb{F}_k$ with $k\geq 3$, we obtain an explicit description of all possible central fibers. For $k\leq 1$, we use the threefold minimal model program to reduce the problem to the case of central fibers whose anticanonical divisor is nef. The remaining surfaces are toric del Pezzo surfaces of degree $8$ with T-singularities. We classify these by studying the birational geometry arising from mutations of Fano polygons. Comments: 47 pages Subjects: Algebraic Geometry (math.AG) MSC classes: 14J26, 14D06 (Primary) 14E30, 14B07, 14M25 (Secondary) Cite as: arXiv:2608.27652 [math.AG] (or arXiv:2608.27652v1 [math.AG] for this version) https://doi.org/10.48550/arXiv.2608.27652 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Juan Pablo Zúñiga [ view email ] [v1] Thu, 27 Aug 2026 19:40:11 UTC (235 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs 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 . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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