Frobenius splitting and geometry of $G$-Schubert varieties

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dc.creatorHe, Xuhua
dc.creatorThomsen, Jesper Funch
dc.date2007-04-05
dc.date2008-09-10
dc.date.accessioned2026-07-07T10:01:31Z
dc.date.available2026-07-07T10:01:31Z
dc.descriptionLet $X$ 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.
dc.descriptionFinal version, 44 pages
dc.identifierhttps://arxiv.org/abs/0704.0778
dc.identifierhttp://arxiv.org/abs/0704.0778
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/168655
dc.subjectAlgebraic Geometry
dc.subjectCommutative Algebra
dc.subjectRepresentation Theory
dc.titleFrobenius splitting and geometry of $G$-Schubert varieties
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