EMBEDDINGS OF HOMOGENEOUS SPACES IN PRIME CHARACTERISTICS

Let G be a reductive linear algebraic group. The simplest example of a projective homogeneous G-variety in characteristic p, not isomorphic to a Bag variety, is the divisor x(0)y(0)(p)+x(1)y(1)(p)+x(2)2)(p) = 0 in P-2 x P-2, which is SL(3) modulo a nonreduced stabilizer containin g the upper triangular matrices. In this paper embeddings of projectiv e homogeneous spaces viewed as G/H, where H is any sub-group scheme co ntaining a Borel subgroup, are studied. We prove that G/H can be ident ified with the orbit of the highest weight line in the projective spac e over the simple G-representation L(lambda) of a certain highest weig ht lambda. This leads to some strange embeddings especially in charact eristic 2, where we give an example in the C-4-case lying on the bound ary of Hartshorne's conjecture on complete intersections. Finally we p rove that ample line bundles on G/H are very ample. This gives a count erexample to Kodaira type vanishing with a very ample line bundle, ans wering an old question of Raynaud.