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 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.