Staircase Patterns in Hirzebruch Surfaces

TL;DR

This paper investigates the structure of staircase phenomena in the ellipsoidal capacity functions of symplectic Hirzebruch surfaces, revealing a complex interplay of obstructions, special parameters, and irrational accumulation points.

Contribution

It provides an almost complete description of staircases in Hirzebruch surfaces, introducing a new approach using almost toric fibrations and analyzing the structure of obstructions and special parameters.

Findings

Most parameters have no staircase due to dense obstructions.

Uncountably many parameters admit staircases with irrational accumulation points.

Special rational parameters are closely related to symmetries and do not admit ascending staircases.

Abstract

The ellipsoidal capacity function of a symplectic four manifold $X$ measures how much the form on $X$ must be dilated in order for it to admit an embedded ellipsoid of eccentricity $z$. In most cases there are just finitely many obstructions to such an embedding besides the volume. If there are infinitely many obstructions, $X$ is said to have a staircase. This paper gives an almost complete description of the staircases in the ellipsoidal capacity functions of the symplectic Hirzebruch surfaces $H_b$ formed by blowing up the projective plane with weight $b$. We describe an interweaving, recursively defined, family of obstructions to symplectic embeddings of ellipsoids that show there is an open dense set of shape parameters $b$ that are blocked, i.e. have no staircase, and an uncountable number of other values of $b$ that do admit staircases. The remaining $b$-values form a countable sequence of special rational numbers that are closely related to the symmetries discussed in Magill--McDuff (arXiv:2106.09143). We show that none of them admit ascending staircases. Conjecturally, none admit descending staircases. Finally, we show that, as long as $b$ is not one of these special rational values, any staircase in $H_b$ has irrational accumulation point. A crucial ingredient of our proofs is the new, more indirect approach to using almost toric fibrations in the analysis of staircases by Magill (arXiv:2204.12460). In particular, the structure of the relevant mutations of the set of almost toric fibrations on $H_b$ is echoed in the structure of the set of blocked $b$-intervals.