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 flag variety, is the divisor $x_0 y_0^p+x_1 y_1^p+x_2 y_2^p=0$ in $P^2\times P^2$, which is $SL_3$ modulo a non-reduced 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 $L(\lambda)$ of a certain highest weight $\lambda$. 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.