Embeddings of Homogeneous Spaces in Prime Characteristics

Let X be a projective algebraic variety over an algebraically closed field k admitting a homogeneous action of a semisimple linear algebraic group G. Then X can be canonically identified with the homogeneous space G/Gx, where x is a closed point in X and Gx the stabilizer group scheme of x. A group scheme over a field of characteristic 0 is reduced so in this case, X is isomorphic to a generalized flag variety G/P , where P is a parabolic subgroup. In [2][7][6][5] the geometry of X in prime characteristic has been studied and it has been shown that a lot of strange phenomena occur when Gx is non-reduced. The simplest example of a projective homogeneous G-space (for G = SL3(k), char k = p > 0) not isomorphic to a generalized flag variety is the divisor x0 y p 0 + x1 y p 1 + x2 y p 2 = 0 in P 2 × P. Since projective homogeneous spaces with non-reduced stabilizers are quite algebraic by construction, we give in §2 of this paper a simple geometric approach for their construction, involving only scheme theoretic images under partial Frobenius morphisms. We choose to do this focusing on the “unseparated incidence variety”. In this case the geometric approach completely determines the cohomology of effective line bundles. The reader unfamiliar with the general concept of projective homogeneous spaces in prime characteristic might find this section useful. The main topic of this paper is the study of embeddings of homogeneous projective spaces. Let X be a projective homogeneous G-space. In §3.2 we show that X can be realized as the G-orbit of the B-stable line in P(L(λ)), where L(λ) denotes the simple G-representation of a certain highest weight λ. This approach leads in §3.3 to examples of some strange embeddings of homogeneous spaces in characteristic 2 one lying on the boundary of Hartshorne’s conjecture on complete intersections (a socalled Hartshorne variety in the terms of [8]). Recall that an ample line bundle L on an algebraic variety X is called normally generated [10] if the multiplication map H(X,L) → H(X,L) is surjective for all n ≥ 1. For generalized flag varieties the very ampleness of ample line bundles follows from normal generation of ample line bundles [1][9] (In [10] one can find a nice and short proof of the fact that an ample normally generated line bundle L is very ample). In the general setting of projective homogeneous spaces, normal generation of ample line bundles is an open question. In §4 we compute the line bundles on projective homogeneous spaces and in §4.2 we use a simple “diagonal” construction to prove that ample line bundles on projective homogeneous Gspaces are very ample. In view of [5] this proves the existence of a counterexample to Kodaira