Comparison of two constructions of noncommutative surfaces with exceptional collections of length 4

Abstract: Recently the Euler forms on numerical Grothendieck groups of rank 4 whose properties mimick that of the Euler form of a smooth projective surface have been classified. This classification depends on a natural number $m$, and suggests the existence of noncommutative surfaces which up to that point had not been considered for $m\geq 2$. These have been constructed for $m=2$ using noncommutative $\mathbb{P}^1$-bundles, and for all $m\geq 2$ by a different construction using maximal orders on $\mathrm{Bl}_x\mathbb{P}^2$. In this article we compare the constructions for $m=2$, i.e. we compare the categories arising from half-ruled del Pezzo quaternion orders on $\mathbb{F}_1$ with noncommutative $\mathbb{P}^1$-bundles on $\mathbb{P}^1$. This can be seen as a noncommutative instance of the classical isomorphism $\mathbb{F}_1\cong\mathrm{Bl}_x\mathbb{P}^2$.