Related papers: On the smooth locus of affine Schubert varieties

Let $\mathscr{G}$ be a special parahoric group scheme of twisted type over the ring of formal power series over $\mathbb{C}$, excluding the absolutely special case of $A_{2\ell}^{(2)}$. Using the methods and results of Zhu, we prove a…

This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type $\mathrm{A}^{(1)}_\ell$). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also…

We give a complete list of smooth and rationally smooth normalized Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat-Tits building. The particular case of the…

We classify all normal Schubert varieties in the affine Grassmannian of a semisimple group over an arbitrary field with special attention to small positive characteristic. The proof is elementary and relies on tangent space calculations for…

It is well known that many geometric properties of Schubert varieties of type $A$ can be interpreted combinatorially. Given two permutations $w,x\in S_n$ we give a combinatorial consequence of the property that the smooth locus of the…

We show that every smooth Schubert variety of affine type $\tilde{A}$ is an iterated fibre bundle of Grassmannians, extending an analogous result by Ryan and Wolper for Schubert varieties of finite type $A$. As a consequence, we finish a…

We study spherical Schubert varieties in the affine Grassmannian. These Schubert varieties have a natural conjectural modular description due to Finkelberg-Mirkovi\'c. This modular description is easily seen to be set-theoretically correct,…

Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical…

Following Contou-Carrere [CC], we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the Tits building associated to the situation. We prove that the rational smoothness of a Schubert variety can be…

In this paper, we consider the GIT quotients of Schubert varieties for the action of a maximal torus. We describe the minuscule Schubert varieties for which the semistable locus is contained in the smooth locus. As a consequence, we study…

We prove in type A a conjecture which describes the ideal of transversal slices to spherical Schubert varieties in the affine Grassmannian. As a corollary, we prove a modular description (due to Finkelberg-Mirkovi\'c) of the spherical…

Let X = G/P be a cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) I give a uniform description (that is, independent of type) of the irreducible components of the…

We prove a criterion for the normality of Schubert varieties in twisted affine Grassmannians in terms of the order of the algebraic fundamental group of a certain Levi subgroup, in particular in small positive characteristic. As an…

A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fibre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showing that a Schubert variety in a generalized…

We prove the Lipman-Zariski conjecture for complex surface singularities of genus one, and also for those of genus two whose link is not a rational homology sphere. As an application, we characterize complex $2$-tori as the only normal…

In this article, we describe explicitely the Gorenstein locus of all minuscule Schubert varieties. This proves a special case of a conjecture of A. Woo and A. Yong (see math.AG/0603273) on the Gorenstein locus of Schubert varieties.

We give an explicit combinatorial description of the irreducible components of the singular locus of a Schubert variety in a flag variety of type A. This implies a conjecture of Lakshmibai and Sandhya.

A smooth projective variety with an action of a torus admits a cell decomposition, called the Bialynicki-Birula decomposition. Singularities of the closures of these cells are not well-known. One of the examples of such closures is a…

For the toric variety X associated to the Bruhat poset of Schubert varieties in the Grassmannian, we describe the singular locus in terms of the faces of the associated polyhedral cone. We also determine the tangent cones at the maximal…

Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri. This result was extended to the other classical types by Lakshmibai. We give a uniform characterization of tangent spaces to Schubert varieties in…