1995/02/16 by Niels Lauritzen, Lauritzen, Niels
Mathematics · #Advanced Topics in Algebra #Rings, Modules, and Algebras #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.alg-geom/9502016
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 x0 y0p+x1 y1p+x2 y2p=0 in P2× P2, which is SL3 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(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.