Ruled surfaces II: toric aspects

The Hirzebruch surfaces – geometrically ruled surfaces over $\mathbb{P}^1$ – are very classical and well-studied objects that in addition turn out to be toric. I suppose that an intuitive reason for why this is the case is that there is at least a 2-torus action on such a surface: $\text{PGL}_2(\mathbb{C})$ acts on both the base and on each fibre and contains a 1-torus as the classes of diagonal matrices.

Apparently they were first studied as embedded surfaces known as ‘scrolls’ depending on two parameters, which can be symplectified into a moment polytope and then dualised to give a toric fan, causing one to realise that only one parameter was needed after all to determine the surface as an abstract variety.

The elementary transformations that move one between the different Hirzebruch surfaces have a toric incarnation too. They also exhibit the main combinatorial features of intersection theory for toric varieties, by which one can distinguish the different Hirzebruch surfaces. They also carry the combinatorics associated to toric projective bundles; both of these are topics I plan to discuss in more detail soon. In the meantime, here are my notes.