American Journal of Mathematics

Let G be a reductive linear algebraic group. The simplest example of a projective homogeneous G-variety in characteristic p, not isomorphic to a flag variety, is the divisor x₀y^p₀ +x₁y^p₁ +x₂y^p₂ = 0 in P² × P², which is SL₃ modulo a nonreduced stabilizer containing the upper triangular matrices. In this paper embeddings of projective homogeneous spaces viewed as G/H, where H is any subgroup scheme containing a Borel subgroup, are studied. We prove that G/H can be identified with the orbit of the highest weight line in the projective space over the simple G-representation L(λ) of a certain highest weight λ. This leads to some strange embeddings especially in characteristic 2, where we give an example in the C₄-case lying on the boundary of Hartshorne's conjecture on complete intersections. Finally we prove that ample line bundles on G/H are very ample. This gives a counterexample to Kodaira type vanishing with a very ample line bundle, answering an old question of Raynaud.