Weighted projective planes and threefold singularities

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

Abstract

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