Unobstructed embeddings in Hirzebruch surfaces

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 blowup. Cristofaro–Gardiner, et al. $\href{https://doi.org/10.48550/arXiv.2004.13062}{\textrm{arXiv:2004.13062}}$ 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 $\href{ https://ui.adsabs.harvard.edu/link_gateway/2022arXiv220306453M/doi:10.48550/arXiv.2203.06453}{\textrm{(arXiv:2203.06453)}}$ to show that these classes are exceptional and that these $b$-values do have infinite staircases.

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Supported by NSF Graduate Research Grant DGE-1650441.

Received 18 May 2022

Received revised 12 June 2023

Accepted 21 June 2023

Published 19 August 2024