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Embeddings of homogeneous spaces in prime characteristics. (English) Zbl 0871.14039
Summary: 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 2 × 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.
MSC:
14M17Homogeneous spaces and generalizations
14G15Finite ground fields
14E25Embeddings (algebraic varieties)