Staircase symmetries in Hirzebruch surfaces

Magill, Nicki; McDuff, Dusa

doi:10.2140/agt.2023.23.4235

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Staircase symmetries in Hirzebruch surfaces

Nicki Magill and Dusa McDuff

Algebraic & Geometric Topology 23 (2023) 4235–4307

DOI: 10.2140/agt.2023.23.4235

Abstract

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).

Keywords

symplectic embeddings in four dimensions, symplectic capacity function, Diophantine equation

Mathematical Subject Classification

Primary: 53D05

Secondary: 11D99

References

Publication

Received: 1 August 2021

Revised: 14 June 2022

Accepted: 26 June 2022

Published: 23 November 2023

Authors

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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Volume 23, issue 9 (2023)