Frobenius splitting and geometry of $G$-Schubert varieties

Xuhua He; Jesper Funch Thomsen

Home/Archive/arXiv

arXiv · 0704.0778

Frobenius splitting and geometry of $G$-Schubert varieties

Xuhua He·Jesper Funch Thomsen

View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.AG math.AC math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Xuhua He, Jesper Funch Thomsen. 2008-09-10. Frobenius splitting and geometry of $G$-Schubert varieties. https://arxiv.org/abs/0704.0778

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Lawson Homology for projective varieties with C^*-action

We establish a decomposition of integral Lawson homology compatible with cycle class maps for projective varieties filtered by Zariski locally trivial affine space bundles over smooth projective bases. Resolving the graph closures produces liftings without a flatness assumption on the graph projections. We prove that these liftings are independent of the chosen resolutions and yield compatible projectors and filtrations on Lawson homology and singular homology. When the bases have dimension at most two, the comparison maps are injective, are isomorphisms in degrees $k>2p$, and have torsion free diagonal cokernels determined by the surface bases. In this case all integral Hodge homology classes are algebraic. For smooth projective varieties with a multiplicative group action and fixed components of dimension at most two, the ranks of the diagonal cokernels form a symmetric polynomial. We also describe the change of this polynomial under blow-ups and compute it for rational varieties obtained by blowing up quartic surfaces. We recover the known smooth motivic decomposition and distinguish its consequences from these refinements. Applications include cones, singular hypersurfaces, toric varieties, finite quotients, symmetric products, and Hilbert schemes of points on surfaces with integral Tate motives.

math.AG↗

arXiv2026

Valuations and henselization

We study the extension of valuations centered in a local domain to its henseliza-tion. We prove that a valuation $ν$ centered in a local domain R uniquely determines a minimal prime H($ν$) of the henselization R h of R and an extension of $ν$ centered in R h /H($ν$), which has the same value group as $ν$. Our method, which assumes neither that R is noetherian nor that it is integrally closed, is to reduce the problem to the extension of the valuation to a quotient of a standard {é}tale local R-algebra and in that situation to draw valuative consequences from the observation that the Newton-Hensel algorithm for constructing roots of polynomials produces sequences that are always pseudo-convergent in the sense of Ostrowski. We then apply this method to the study of the approximation of elements of the henselization of a valued field by elements of the field and give a characterization of the henselian property of a local domain (R, m R) in terms of the limits of certain pseudo-convergent sequences of elements of m R for a valuation centered in it. Another consequence of our work is to establish in full generality a bijective correspondence between the minimal primes of the henselization of a local domain R and the connected components of the Riemann-Zariski space of valuations centered in R.

math.AG↗

arXiv2026

Algebra of global sections of $ψ$-bundles on $\bar{M}_{0,n}$

We consider the ${\mathbb Z}^n$-graded algebra of global sections of line bundles generated by the standard line bundles $L_1,\ldots,L_n$ on $\bar{M}_{0,n}$. We find a simple presentation of this algebra by generators and quadratic relations. As an application we prove that the moduli space $\bar{M}_{0,n}[ψ]$ of $ψ$-stable curves of genus $0$ is Cohen-Macaulay and normal, and the natural map $\bar{M}_{0,n}\to \bar{M}_{0,n}[ψ]$ is a rational resolution.

math.AG↗