Alexander Woo
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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We answer some questions related to multiplicity formulas by Rosenthal and Zelevinsky and by Lakshmibai and Weyman for points on Schubert varieties in Grassmannians. In particular, we give combinatorial interpretations in terms of nonintersecting lattice paths of these formulas, which makes the equality of the two formulas immediately obvious. Furthermore we provide an alternative determinantal formula for these multiplicities, and we show that they count semistandard tableaux of unusual shapes.
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 obtain an explicit determinantal formula for the multiplicity of any point on a classical Schubert variety.
Recently, Lauritzen, Raben-Pedersen and Thomsen proved that Schubert varieties are globally $F$-regular. We give another proof.
We prove a conjecture by Kreiman and Lakshmibai on a combinatorial description of multiplicities of points on Schubert varieties in Graszmannians in terms of certain sets of reflections in the corresponding Weyl group. The proof is accomplished by setting up a bijection between these sets of reflections and the author's previous combinatorial interpretation of these multiplicities in terms of nonintersecting lattice paths (Séminaire Lotharingien Combin. 45 (2001), Article B45c; see http://www.mat.univie.ac.at/~slc/wpapers/s45kratt.html or http://www.arxiv.org/abs/math.AG/0011129).
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.
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.
In this paper we describe the relationship between the finite free resolutions of perfect ideals in split format (for Dynkin formats) and certain intersections of opposite Schubert varieties with the big cell for homogeneous spaces $G/P$ where $P$ is a maximal parabolic subgroup.
We study Hilbert-Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Groebner degenerations of the Kazhdan-Lusztig ideal. In the covexillary case, we give a positive combinatorial rule for multiplicity by establishing (with a Groebner basis) a reduced and equidimensional limit whose Stanley-Reisner simplicial complex is homeomorphic to a shellable ball or sphere. We show that multiplicity counts the number of facets of this complex. We also obtain a formula for the Hilbert series of the local ring. In particular, our work gives a multiplicity rule for Grassman…
This note computes a Gröbner basis for the ideal defining a union of Schubert varieties. More precisely, it computes a Gröbner basis for unions of schemes given by northwest rank conditions on the space of all matrices of a fixed size. Schemes given by northwest rank conditions include classical determinantal varieties and matrix Schubert varieties--closures of Schubert varieties lifted from the flag manifold to the space of matrices.
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…
Minimal equations for matrix Schubert varieties
本文确定了富尔顿舒伯特行列式理想的显式极小生成元，并讨论了相关推论。
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.
Knutson and Miller (2005) established a connection between the anti-diagonal Gröbner degenerations of matrix Schubert varieties and the pre-existing combinatorics of pipe dreams. They used this correspondence to give a geometrically-natural explanation for the appearance of the combinatorially-defined Schubert polynomials as representatives of Schubert classes. Recently, Hamaker, Pechenik, and Weigandt (2022) proposed a similar connection between diagonal degenerations of matrix Schubert varieties and bumpless pipe dreams, newer combinatorial objects introduced by Lam, Lee, and Shimozono (2021…
We study exceptional minuscule Schubert varieties and provide the defining equations of the defining ideals of their intersection with the big open cell. We also provide the resolutions of these ideals and characterize some of them in terms of fundamental examples of ideals in the theory of Gorenstein ideals.
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, proj…
We prove a recent conjecture of Kreiman and Lakshmibai about Hilbert functions of points on Schubert varieties in Grassmannians.
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.
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 …
Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo-Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo-Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo-Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo-Mumfor…
Residual Intersections and Schubert Varieties
本文提出一种对 ADE 型一致的表示论模式，证明某些相对 Schubert 簇的理想可由两个几何链接的相对 Schubert 簇的剩余交得到，并指出其自由消解对有限自由消解结构的重要性。
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.
We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore Stanley-Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths.
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.
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。
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.
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。
We show how the Frobenius splitting method of Mehta-Ramanathan implies the Koszul property of projective coordinate rings of Schubert varieties.
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…
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.
Algebra and geometry of ASM weak order
该研究将ASM上的弱布鲁哈特序与代数几何结合，刻画了ASM簇的余维数，证明弱序算子与K理论差分算子可交换且满足特定导数公式，并将结果推广至矩阵舒伯特簇的任意并。
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.
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.
In [MV], some correspondences were defined between critical points of master functions associated to sl_{N+1} and subspaces of C[x] with given ramification properties. In this paper we show that these correspondences are in fact scheme theoretic isomorphisms of appropriate schemes. This gives relations between multiplicities of critical point loci of the relevant master functions and multiplicities in Schubert calculus.
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。
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].
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.
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…
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.
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。
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 …
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).
Fine multidegrees, universal Grobner bases, and matrix Schubert varieties
本文给出了多项式集合成为理想通用Gröbner基的精细多度判据，并引入精细Schubert多项式计算矩阵Schubert簇在(ℙ¹)^N中闭包的精细多度，进而为某些置换的矩阵Schubert簇理想构造通用Gröbner基。
From Schubert Varieties to Doubly-Spherical Varieties
研究刻画 horospherical Schubert 簇，证明其点稳定子群为强可解代数群并讨论连通性，同时提出双球面簇概念，证明近环面 Schubert 簇均为双球面簇。
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 $G/P$.
Nearly Toric Schubert Varieties of Type A
本文引入近似环面簇概念，研究A型Schubert簇中的相应例子，给出光滑与奇异情形的组合刻画、计数及生成级数，并联系Dyck路径。
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.
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