N. Shepherd-Barron

Recognising flag varieties and reductive groups

Fix a flat and projective morphism X→Σ$X\rightarrow \Sigma$ of schemes. We show, first, that any set of P1${\mathbb {P}}^1$ ‐fibrations on X$X$ defines a set of simple roots, a set of simple coroots and a Cartan matrix C$C$ . Second, X$X$ is an étale F${\mathcal {F}}$ ‐bundle over some projective Σ$\Sigma$ ‐scheme, whe...