Embeddings of homogeneous spaces in prime characteristics

Source abstract

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 x 0 y p 0 + x 1 y p 1 + x 2 y p 2 = 0 in P 2 × P 2 , which is SL 3 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 C 4 -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.