齐性空间在素特征中的嵌入
Embeddings of homogeneous spaces in prime characteristics
摘要：
设$G$是一个约化线性代数群.在特征$p$中的投影齐次$G$ -簇的最简单的例子，不同构于标志簇，是$P^2\times P^2$中的除数$x_0 y_0^p+x_1 y_1^p+x_2 y_2^p=0$，它是$SL_3$模包含上三角矩阵的非约化稳定子。本文研究了投射齐性空间$G/H$的嵌入问题，其中$H$是包含Borel子群的子群概型.我们证明了$G/H$可以被确定为具有某个最高权$λ$的简单$G$ -表示$L(λ)$上的射影空间中的最高权线的轨道。这导致了一些奇怪的嵌入，特别是在特征$2$，在那里我们给一个例子中的$C_4$ -情况下躺在边界上的Hartshorne的猜想完全相交。最后证明了$G/H$上的充分线丛是非常充分的。这给出了Kodaira型消失的一个反例，回答了雷诺的一个老问题。
英文摘要：
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 $L(λ)$ 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.
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