摘要翻译：
设$X$是正特征代数闭域$K$上连通约化群$G$的等变嵌入。设$B$表示$G$的Borel子群。$X$中的$G$-Schubert变体是$\diag(G)\cdot V$形式的子变体，其中$V$是$X$中的$B\乘以B$-Orbit闭包。在$x$是一组伴随类型的奇妙压缩的情况下，$G$-Schubert变体是Lusztig的$G$-稳定块的闭包。我们证明了$X$允许Frobenius分裂，它与所有$G$-Schubert变体兼容。此外，当$X$是光滑的、射影的和环形的时，则$X$中的任何$G$-Schubert类都允许一个稳定的Frobenius分裂。虽然这表明$G$-Schubert变体有很好的奇点，但我们给出了一个非正常的$G$-Schubert变体的例子，在一组$G_2$型的奇妙紧致中。最后，我们还将Frobenius分裂的结果推广到更一般的$\MathcalR$-Schubert类。
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英文标题：
《Frobenius splitting and geometry of $G$-Schubert varieties》
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作者：
Xuhua He and Jesper Funch Thomsen
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Commutative Algebra        交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
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一级分类：Mathematics        数学
二级分类：Representation Theory        表象理论
分类描述：Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra
代数和群的线性表示，李理论，结合代数，多重线性代数
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英文摘要：
  Let $X$ be an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic. Let $B$ denote a Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the form $\diag(G) \cdot V$, where $V$ is a $B \times B$-orbit closure in $X$. In the case where $X$ is the wonderful compactification of a group of adjoint type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable pieces. We prove that $X$ admits a Frobenius splitting which is compatible with all $G$-Schubert varieties. Moreover, when $X$ is smooth, projective and toroidal, then any $G$-Schubert variety in $X$ admits a stable Frobenius splitting along an ample divisors. Although this indicates that $G$-Schubert varieties have nice singularities we present an example of a non-normal $G$-Schubert variety in the wonderful compactification of a group of type $G_2$. Finally we also extend the Frobenius splitting results to the more general class of $\mathcal R$-Schubert varieties.
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PDF链接：
https://arxiv.org/pdf/0704.0778