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Staircase symmetries in Hirzebruch surfaces

Nicki Magill and Dusa McDuff

Algebraic & Geometric Topology 23 (2023) 4235–4307

This paper continues the investigation of staircases in the family of Hirzebruch surfaces formed by blowing up the projective plane with weight b, that was started by Bertozzi, Holm Maw, McDuff, Mwakyoma, Pires and Weiler (2021). We explain the symmetries underlying the structure of the set of b that admit staircases, and show how the properties of these symmetries arise from a governing Diophantine equation. We also greatly simplify the techniques needed to show that a family of steps does form a staircase by using arithmetic properties of the accumulation function. There should be analogous results about both staircases and mutations for the other rational toric domains considered, for example, by Cristofaro-Gardiner, Holm, Mandini and Pires (2020) and by Casals and Vianna (2022).

symplectic embeddings in four dimensions, symplectic capacity function, Diophantine equation
Mathematical Subject Classification
Primary: 53D05
Secondary: 11D99
Received: 1 August 2021
Revised: 14 June 2022
Accepted: 26 June 2022
Published: 23 November 2023
Nicki Magill
Mathematics Department
Cornell University
Ithaca, NY
United States
Dusa McDuff
Mathematics Department
Barnard College
New York, NY
United States

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