Steven V Sam, Jerzy Weyman
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.
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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.
Recently, Lauritzen, Raben-Pedersen and Thomsen proved that Schubert varieties are globally $F$-regular. We give another proof.
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 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.
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.
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。
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…
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。
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 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.
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。
We study Gröbner degenerations of Schubert varieties inside flag varieties. We consider toric degenerations of flag varieties induced by matching fields and semi-standard Young tableaux. We describe an analogue of matching field ideals for Schubert varieties inside the flag variety and give a complete characterization of toric ideals among them. We use a combinatorial approach to standard monomial theory to show that block diagonal matching fields give rise to toric degenerations. Our methods and results use the combinatorics of permutations associated to Schubert varieties, matching fields an…
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…
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。
Matrix Schubert varieties are the closures of the orbits of $B\times B$ acting on all $n\times n$ matrices, where $B$ is the group of invertible lower triangular matrices. Extending work of Fulton, Knutson and Miller identified a Gröbner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many othe…
LS Algebras, Valuations and Schubert Varieties
本文借助 LS 代数提出代数方法，将标准单项式理论、Newton-Okounkov 体与赋值联系起来，并在 Schubert 簇中证明其与 Seshadri 分层方法相容。
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 study the combinatorics of Gröbner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by tableaux induced by matching fields in the sense of Sturmfels and Zelevinsky. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Plücker algebra. This leads to a new family of toric degenerations of Grassmannians. Moreover, we apply our results to construct a family of Gröbner degenerations of Schubert varieties inside Grassmannians. We provide a complete characterizatio…
We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of cert…
Antidiagonal Initial Complexes of Infinite Matrix Schubert Varieties are Cohen-Macaulay
本文证明了无限矩阵Schubert变种的反 diagonals初始复形在特定条件下是Cohen-Macaulay的，从而提供了新的非Noetherian Cohen-Macaulay环的例子。
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