Staircase symmetries in Hirzebruch surfaces

TL;DR

This paper explores the symmetries and underlying Diophantine equations governing staircases in Hirzebruch surfaces, simplifying techniques to identify staircase structures and suggesting broader applicability to other rational toric domains.

Contribution

It reveals the symmetry structures and simplifies the detection of staircase formations in Hirzebruch surfaces using arithmetic properties, extending understanding of these geometric phenomena.

Findings

Identified symmetries in the set of parameters admitting staircases.

Connected staircase properties to a governing Diophantine equation.

Simplified techniques for proving staircase formations using arithmetic properties.

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 in Bertozzi, Holm et al. in arXiv:2010.08567. 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 et al. in arXiv:2004.07829 and by Casals--Vianna in arXiv:2004.13232.