J. Kuttler, V. Lakshmibai
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 determine explicitly the tangent spaces at singular points. For a general affine Schubert variety, we give partial results on the singular locus.
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Our concern in this paper is the dimension and inclusion relations of Schubert varieties in twisted partial affine flag varieties. In the end we apply our results to some local models of certain Schubert varieties.
A permutation is called covexillary if it avoids the pattern $3412$. We construct an open embedding of a covexillary matrix Schubert variety into a Grassmannian Schubert variety. As applications of this embedding, we show that the characteristic cycles of covexillary Schubert varieties are irreducible, and provide a new proof of Lascoux's model computing Kazhdan-Lusztig polynomials of vexillary permutations. Combining the above embedding with earlier work of the author on the conormal varieties of Grassmannian Schubert varieties, we develop an algebraic criterion identifying the conormal varie…
In this work we extend some previously known results on the automorphism group of Schubert varieties. We consider the Schubert conditions which define a Schubert variety. An automorphism of the Grassmannian fixes a Schubert variety contained in it if and only if it fixes certain elements of the flag used to define the Schubert conditions.
We prove a recent conjecture of Kreiman and Lakshmibai about Hilbert functions of points on Schubert varieties in Grassmannians.
We study twisted forms of Schubert cells in generalized Severi-Brauer varieties, and show that the codimension $2$ Chow groups of these varieties are torsion free in certain cases, using the topological filtration on their K-theory
We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov-Libgober classes of Schubert varieties in general homogeneous spaces G/P. While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi-Tarasov-Varchenko we prove that …
Rigid Schubert classes in partial flag varieties
本文研究 A 型偏旗簇及 B、D 型正交偏旗簇中 Schubert 类的刚性，并给出判定其刚性的数值条件。
We describe the generic singularity of a Schubert variety of type A on each irreducible component of its singular locus. This singularity is given either by a cone of rank one matrices, or a quadratic cone.
Intersections of Schubert varieties and smooth $T$-stable subvarieties of flag varieties
本文研究了正则半单 Hessenberg 簇中 Bialynicki-Birula 胞腔闭包（即 Hessenberg Schubert 簇）的光滑性，通过考察 Schubert 簇与 Hessenberg 簇交集的不可约性与光滑性，得出了该簇光滑的充分条件。
Let $P$ be a parabolic subgroup in $G=SL_n(\mathbf k)$, for $\mathbf k$ an algebraically closed field. We show that there is a $G$-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle $T^*G/P$. Restricting this identification to the conormal variety $N^*X(w)$ of a Schubert divisor $X(w)$ in $G/P$, we show that there is a compactification of $N^*X(w)$ as an affine Schubert variety. It follows that $N^*X(w)$ is normal, Cohen-Macaulay, and Frobenius split.
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 the smoothness of torus quotients of Schubert varieties in the Grassmannian. We also prove that the torus quotient of any Schubert variety in the homogeneous space $SL(n, \mathbb C)/P$ is projectively normal with respect to the line bundle $\mathcal L_{α_0}$ and the quotient space is a projective space, where the line bundle $\mathcal L_{α_0}$ and the parabol…
Given a Schubert variety X_w, we exhibit a divisor Δ, defined over the integers, such that the pair (X_w,Δ) is log Fano in all characteristics.
Let $G=SL(n, \mathbb{C}),$ and $T$ be a maximal torus of $G,$ where $n$ is a positive even integer. In this article, we study the GIT quotients of the Schubert varieties in the Grassmannian $G_{2,n}.$ We prove that the GIT quotients of the Richardson varieties in the minimal dimensional Schubert variety admitting stable points in $G_{2,n}$ are projective spaces. Further, we prove that the GIT quotients of certain Richardson varieties in $G_{2,n}$ are projective toric varieties. Also, we prove that the GIT quotients of the Schubert varieties in $G_{2,n}$ have at most finite set of singular poin…
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 expressed in terms of a subspace of the Zariski tangent space called, the combinatorial tangent space. For this, we use a characterization of rational smoothness of a Schubert variety introduced by Carrell and Peterson [CP].
Symplectic conditions on Grassmannian, flag, and Schubert varieties
本文用Plücker坐标的线性方程刻画A型簇中C型Grassmann、旗与Schubert簇，并研究其理想生成、局部方程数、完全交性质及非微小情形的光滑性。
This note constructs the flat toric degeneration of the manifold FL_n of flags in C^n from [Gonciulea-Lakshmibai 96] as an explicit GIT quotient of the Gr"obner degeneration in [Knutson-Miller 03]. This implies that Schubert varieties degenerate to reduced unions of toric varieties, associated to faces indexed by rc-graphs (reduced pipe dreams) in the Gel'fand-Cetlin polytope. Our explicit description of the toric degeneration of FL_n provides a simple explanation of how Gel'fand-Cetlin decompositions for irreducible polynomial representations of GL_n arise via geometric quantization.
We use incidence relations running in two directions in order to construct a Kempf-Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott-Samelson resolutions. The second process led to the introduction of W-flag varieties, algebro-geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of v…
Let $\mbox{IG}(k,2n+1)$ be the odd symplectic Grassmannian. It is a quasi-ho\-mo\-ge\-neous space with homogeneous-like behavior. A very limited description of curve neighborhoods of Schubert varieties in $\mbox{IG}(k,2n+1)$ was used by Mihalcea and the second named author to prove an (equivariant) quantum Chevalley rule. In this paper we give a full description of the irreducible components of curve neighborhoods in terms of the Hecke product of (appropriate) Weyl group elements, $k$-strict partitions, and BC-partitions. The latter set of partitions respect the Bruhat order with inclusions. O…
Multi-rigidity of Schubert classes in partial flag varieties
本文研究有理齐性空间中 Schubert 类的多重刚性，刻画 A、B、D 型部分旗簇情形，并由广义 Grassmann 簇及单群情形推出一般 $G/P$ 的刚性结论。
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 singular locus of a Schubert variety Y in X in terms of representation theoretic data. The result is based on a recent characterization of the Schubert varieties by an non-negative integer A and a marked Dynkin diagram. Corollaries include: (1) the variety is smooth if and only if A=0; (2) if G of Type ADE, then the singular locus occurs in codimension at least t…
The polynomial ring $B$ in infinitely many indeterminates $(x_1,x_2,\ldots)$, with rational coefficients, has a vector space basis of Schur polynomials, parametrized by partitions. The goal of this note is to provide an explanation of the following fact. If $\blamb$ is a partition of weight $d$, then the partial derivative of order $d$ with respect to $x_1$ of the Schur polynomial $S_\blamb(\bfx)$ coincides with the Plücker degree of the Schubert variety of dimension $d$ associated to $\blamb$, equal to the number of standard Young tableaux of shape $\blamb$. The generating function encoding a…
In this paper we construct free resolutions of certain class of closed subvarieties of affine space of symmetric matrices (of a given size). Our class covers the symmetric determinantal varieties (i.e., determinantal varieties in the space of symmetric matrices), whose resolutions were first constructed by Józefiak-Pragacz-Weyman. Our approach follows the techniques developed in Free Resolutions of Some Schubert Singularities by Kummini-Lakshmibai-Sastry-Seshadri, and uses the geometry of Schubert varieties.
Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.
Schubert varieties for the super affine Grassmannian of $GL_{n|m}$
研究一般线性超群$GL_{n|m}$的仿射格拉斯曼型的舒伯特簇，通过低维情形（$n|m=1|1$和$2|1$）的显式计算，描述超簇结构、坐标、方程和态射。
We will describe a one-step "Gorensteinization" process for a Schubert variety by blowing-up along its boundary divisor. The local question involves Kazhdan-Lusztig varieties which can be degenerated to affine toric schemes defined using the Stanley-Reisner ideal of a subword complex. The blow-up along the boundary in this toric case is in fact Gorenstein. We show that there exists a degeneration of the blow-up of the Kazhdan-Lusztig variety to this Gorenstein scheme, allowing us to extend this result to Schubert varieties in general. The potential use of this one-step Gorensteinization to des…
We give a closed-form formula for the Hilbert function of the tangent cone at the identity of a Schubert variety X in the Grassmannian in both group theoretic and combinatorial terms. We also give a formula for the multiplicity of X at the identity, and a Grobner basis for the ideal defining the intersection of X with the opposite cell as a closed subvariety of the opposite cell. We give conjectures for the Hilbert function and multiplicity at points other than the identity.
Maximal disjoint Schubert cycles in rational homogeneous varieties
本文通过研究经典类型有理齐次簇的 Chow 环中低余维数有效零因子及有效良好可除性，探讨了这些簇之间非常数映射的存在性，推广了射影空间和 Grassmann 簇的已知结果。
Consider k x n matrices with rank conditions placed on intervals of columns. The ranks that are actually achievable correspond naturally to upper triangular partial permutation matrices, and we call the corresponding subvarieties of Gr(k,n) the _interval positroid varieties_, as this class lies within the class of positroid varieties studied in [Knutson-Lam-Speyer]. It includes Schubert and opposite Schubert varieties, and their intersections, and is Grassmann dual to the projection varieties of [Billey-Coskun]. Vakil's "geometric Littlewood-Richardson rule" [Vakil] uses certain degenerations …
We characterize complete intersection matrix Schubert varieties, generalizing the classical result on one-sided ladder determinantal varieties. We also give a new proof of the F-rationality of matrix Schubert varieties. Although it is known that such varieties are F-regular (hence F-rational) by the global F-regularity of Schubert varieties, our proof is of independent interest since it does not require the Bott-Samelson resolution. As a consequence, this provides an alternative proof of the classical fact that Schubert varieties in flag varieties are normal and have rational singularities.
We consider tangent cones to Schubert subvarieties of the flag variety $G/B$, where $B$ is a Borel subgroup of a reductive complex algebraic group $G$ of type $E_6$, $E_7$ or $E_8$. We prove that if $w_1$ and $w_2$ form a good pair of involutions in the Weyl group $W$ of $G$ then the tangent cones $C_{w_1}$ and $C_{w_2}$ to the corresponding Schubert subvarieties of $G/B$ do not coincide as subschemes of the tangent space to $G/B$ at the neutral point.
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 provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with any Bott-Samelson class in algebraic cobordism of the grassmannian. We also establish some results for generalized Schubert polynomials for hyperbolic formal group laws.
We show how the Frobenius splitting method of Mehta-Ramanathan implies the Koszul property of projective coordinate rings of Schubert varieties.
These are extended notes of a talk given at Maurice Auslander Distinguished Lectures and International Conference (Woods Hole, MA) in April 2013. Their aim is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V.Kiritchenko, V.Timorin and the author on a new approach to Schubert calculus on full flag varieties via combinatorics…
Recently, Lauritzen, Raben-Pedersen and Thomsen proved that Schubert varieties are globally $F$-regular. We give another proof.
Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20-dimensional $F_4$-homogeneous manifold associated to a short simple root.
We obtain an explicit determinantal formula for the multiplicity of any point on a classical Schubert variety.
Gorenstein toric Schubert varieties in Grassmannians
本文研究格拉斯曼流形中的环面Schubert簇，给出了Gorenstein环面Schubert簇的扇的显式描述，并证明其均为Fano簇。
We consider the action of the Levi subgroup of a parabolic subgroup that stabilizes a Schubert variety. We show that a smooth Schubert variety is a homogeneous space for a parabolic subgroup, or it has a smooth Schubert divisor. Further, we show that all smooth Schubert varieties in a (partial) flag variety of a rank two simple algebraic group are spherical.
On the Classification of Schubert Varieties in Partial Flag Varieties
本文在部分旗流形中推广了Schubert流形的分类，通过根系折叠得到等价对，并确定了Schubert三重流形等价类的上界。
Levi subgroup actions on Schubert varieties are studied. In the case of partial flag varieties, the horospherical actions are determined. This leads to a characterization of the toroidal and horospherical partial flag varieties with Picard number 1. In the more general case, we provide a set of necessary conditions for the action of a Levi subgroup on a Schubert variety to be toroidal. The singular locus of a (co)minuscule Schubert variety is shown to contain all the $L_{max}$-stable Schubert subvarieties, where $L_{max}$ is the standard Levi subgroup of the maximal parabolic which acts on the…
Schubert varieties in the full flag variety of Kac-Moody type are indexed by elements of the corresponding Weyl group. We give a practical criterion for when two such Schubert varieties (from potentially different flag varieties) are isomorphic, in terms of the Cartan matrix and reduced words for the indexing Weyl group elements. As a corollary, we show that two such Schubert varieties are isomorphic if and only if there is an isomorphism between their integral cohomology rings that preserves the Schubert basis.
Tropical matroid Schubert varieties and the graded Möbius algebra
本文引入热带Matroid Schubert变种，证明其热带上同调环与Matroid的分级Möbius代数同构，扩展了Arrangement Schubert变种到任意Matroid。
A study of intersections of Schubert varieties
该研究聚焦旗流形中两个舒伯特簇交的局部与整体相交上同调，且在当前版本补充了新的参考文献，为舒伯特簇交的上同调研究提供参考。
We give a criterion for smoothness of a point in any Schubert variety in any G/B in terms of the nil Hecke ring.
We characterize by pattern avoidance the Schubert varieties for GL_n which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighborhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require working with more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov …
After proving that every Schubert variety in the full flag variety of a complex reductive group $G$ is a general Hessenberg variety, we show that not all such Schubert varieties are adjoint Hessenberg varieties. In fact, in types A and C, we provide pattern avoidance criteria implying that the proportion of Schubert varieties that are adjoint Hessenberg varieties approaches zero as the rank of $G$ increases. We show also that in type A, some Schubert varieties are not isomorphic to any adjoint Hessenberg variety.
We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual the Schubert VHS form a basis of the invariant characteristic cohomology associated to the infinitesimal period relation (a.k.a. Griffiths transversality).
We classify all Q-factorializations of (co)minuscule Schubert varieties by using their Mori dream space structure. As a corollary we obtain a description of all IH-small resolutions of (co)minuscule Schubert varieties generalizing results of Perrin. We improve his results by including algebraically closed fields of positive characteristic and cominuscule Schubert varieties. Moreover, the use of Q-factorializations and Mori dream spaces simplifies the arguments substantially.
Normality of Schubert varieties in affine Grassmannians
本文分类了任意域上半单群仿射格拉斯曼流形中的正规Schubert簇，特别关注小正特征，通过切空间计算、Levi引理和极小退化分类给出初等证明。
The Isomorphism Problem for cominuscule Schubert Varieties
本文研究了余微缩旗簇中 Schubert 簇的同构问题，证明了不同簇间的同构等价于其对应标记偏序集的同构，且该证明独立于 Lie 类型。
We describe the effect of Feigin's flat degeneration of the type $\textrm{A}$ flag variety on its Schubert varieties. In particular, we study when they stay irreducible and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some Schubert varieties with Richardson varieties in higher rank partial flag varieties.
We show that the smooth horizontal Schubert subvarieties of a rational homogeneous variety $G/P$ are homogeneously embedded cominuscule $G'/P'$, and are classified by subdiagrams of a Dynkin diagram. This generalizes the classification of smooth Schubert varieties in cominuscule $G/P$.
We enumerate the number of staircase diagrams over classically finite $E$-type Dynkin diagrams, extending the work of Richmond and Slofstra (Staircase Diagrams and Enumeration of smooth Schubert varieties) and completing the enumeration of staircase diagrams over finite type Dynkin diagrams. The staircase diagrams are in bijection to smooth and rationally smooth Schubert varieties over $E$-type thereby giving an enumeration of these varieties.
We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of orthogonal flag varieties. We use these polynomials to describe the arithmetic Schubert calculus on these spaces. We also give a method to compute the natural arithmetic intersection numbers which arise, and prove that they are all rational numbers.
We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank. We deduce that the corresponding degenerate flag varieties are isomorphic to Schubert varieties in any characteristic.
Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of Y. This extends (independent) work of M. Walters, R. Bryant and J. Hong. Key tools include: (i) a new characterization of Schubert varieties that generalizes the well known description of the smooth Schubert varieties by connected sub-diagrams of a Dynkin diagram; and (ii) an algebraic Laplacian (a la Kostant), which is used to anal…
We prove that Schubert varieties are globally F-regular in the sense of Karen Smith. We apply this result to the category of equivariant and holonomic D-modules on flag varieties in positive characteristic. Here recent results of Blickle are shown to imply that the simple D-modules coincide with local cohomology sheaves with support in Schubert varieties. Using a local Grothendieck-Cousin complex we prove that the decomposition of local cohomology sheaves with support in Schubert cells is multiplicity free.
In this article, we consider Schubert codes, linear codes associated to Schubert varieties, and discuss minimum weight codewords for dual Schubert codes. The notion of lines in Schubert varieties is looked closely at, and it has been proved that the supports of the minimum weight codewords of the dual Schubert codes lie on lines and any three points on a line in Schubert variety correspond to the support of some minimum weight parity check for the Schubert code. We use these lines in Schubert varieties to construct orthogonal parity checks for certain Schubert codes and use them for majority l…
We calculate using Macaulay 2 the multiplicities of the most singular point on Schubert varieties on Gl(n)/B for $n=5,6$. The method of computation is described and tables of the results are included.
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