Ruled surfaces I

The title is slightly misleading: I’ve actually been learning about ‘geometrically ruled surfaces’ this week mainly from some lecture notes written by Vakil. The motivation partly came from Gross-Hacking-Keel, where certain ‘elementary transformations’ provide the appropriate notion of mutation in their presentation of cluster varieties. These were classically studied by Hirzebruch among others first in the situation of \mathbb{P}^1-bundles over curves.

A geometrically ruled surface is a surface X\to C over a curve whose fibres are all rational curves (equivalently, the generic fibre is rational). A ready source of these is \mathbb{P}^1-bundles over curves, which turns out to be all of them (the Noether-Enriques theorem). Like projective bundles in general, one can perform very explicit and comprehensive calculations of their intersection theory. This then allows one to distinguish different geometrically ruled surfaces over the same base; for instance the Hirzebruch surfaces

\mathbb{F}_n:=\mathbb{P}\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n)

over \mathbb{P}^1. These are the only Fano geometrically ruled surfaces, and are also toric. They are also known as ‘scrolls’, which is maybe a more romantic name befitting something that feels so classical.

Classifying projective bundles is a rather more modern activity: these are locally trivial fibrations by projective spaces, or \text{PGL}_n-torsors, which then are classified by H^1(\text{PGL}_n). In the case when the base is a curve, it’s immediate to extract from this the classification of projective bundles as projectivisations of vector bundles modulo the action of the Picard group by tensoring, about which much of the computation pivots.

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