By a surprising construction due to Chevalley, simple algebraic groups and their root systems are defined over the integers. Over algebraically closed fields,
one might thus think that the geometry of homogeneous varieties does not depend much on the characteristic. However, this is far from being true: in positive
characteristics, the behavior is often more pathological (if one is pessimistic) or richer (if one is optimistic). In both cases, it can be useful to keep in
mind a few examples; among them, we will introduce infinitesimal groups, non-Fano rational homogeneous varieties, and very special isogenies.