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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 $x_0y^p_0 +x_1y_1^p +x_2y^p_2 =0$ in $\bbfP^2 \times \bbfP^2$, 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 $L (\lambda)$ 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.

14M17Homogeneous spaces and generalizations
14G15Finite ground fields
14E25Embeddings (algebraic varieties)
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