Unobstructed embeddings in Hirzebruch surfaces

TL;DR

This paper investigates the structure of the ellipsoid embedding function in symplectic Hirzebruch surfaces, constructing full-fillings at accumulation points for specific irrational parameters, and linking these to infinite staircase phenomena.

Contribution

It introduces a method using almost toric fibrations to identify potential staircase values in Hirzebruch surfaces, connecting obstructive classes with mutation sequences.

Findings

Constructed full-fillings at accumulation points for irrational b-values

Identified a recursive structure linking obstructive classes and mutations

Confirmed the existence of infinite staircases for certain b-values

Abstract

This paper continues the study of the ellipsoid embedding function of symplectic Hirzebruch surfaces parametrized by $b \in (0,1)$, the size of the symplectic blow-up. Cristofaro-Gardiner, et al. (arxiv: 2004.13062) found that if the embedding function for a Hirzebruch surface has an infinite staircase, then the function is equal to the volume curve at the accumulation point of the staircase. Here, we use almost toric fibrations to construct full-fillings at the accumulation points for an infinite family of recursively defined irrational $b$-values implying these $b$ are potential staircase values. The $b$-values are defined via a family of obstructive classes defined in Magill-McDuff-Weiler (arxiv:2203.06453). There is a correspondence between the recursive, interwoven structure of the obstructive classes and the sequence of possible mutations in the almost toric fibrations. This result is used in Magill-McDuff-Weiler (arxiv:2203.06453) to show that these classes are exceptional and that these $b$-values do have infinite staircases.