Weighted projective planes and threefold singularities

Kelleher, Sarah (2022) Weighted projective planes and threefold singularities. PhD thesis, University of Glasgow.

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This thesis studies weighted projective planes and their connection to threefold singularities. In particular, we study the Veronese subring S→𝑥 of the ring S associated with the weighted projective plane 𝕏 for choices of →𝑥 in the grading group 𝕃. We show that there exists a projective, birational map T→𝑥 ⟶ Spec S→𝑥 under mild restrictions on →𝑥. We then show that when →𝑥 = -→ω, the dualising element, this map is a blow-up. In the toric setting, we show that in certain situations the singularities of S-→ω can be identified with the familiar cyclic quotient singularities and the map T-→ω ⟶ Spec S-→ω is a weighted blow-up. In particular, it is a crepant map. We also construct a tilting object on T-→ω in this setting. Away from the toric setting, we are able to construct tilting objects in some instances and we study some examples in depth to construct a full resolution and identify noncommutative resolutions of these singularities.

Item Type: Thesis (PhD)
Qualification Level: Doctoral
Additional Information: Supported by funding from the Engineering and Physical Sciences Research Council (EPSRC).
Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
Supervisor's Name: Bellamy, Professor Gwyn and Wemyss, Professor Michael
Date of Award: 2022
Depositing User: Theses Team
Unique ID: glathesis:2022-83074
Copyright: Copyright of this thesis is held by the author.
Date Deposited: 10 Aug 2022 13:40
Last Modified: 10 Aug 2022 13:40
Thesis DOI: 10.5525/gla.thesis.83074
URI: https://theses.gla.ac.uk/id/eprint/83074

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