Questions tagged [schubert-calculus]

Schubert calculus is the study of flag varieties, which are quotients of algebraic groups (usually complex semisimple, but sometimes over the real numbers or even finite fields) by parabolic subgroups.

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Chern class of bundle of linear functions on $G(2,4)$

Let $G=G(1,3)$ be the grassmanian of lines in $\mathbb{C}P^3$. Let $E$ be the bundle on $G$ whose fiber over $\ell$ are homogeneous linear functions on $\ell$. I am interested in $c(E)=1+c_1(E)+c_2(E)$...
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Algorithm for expressing a homogeneous polynomial with integer coefficients nonnegatively (if possible) in terms of specific dependent binomials

I have specific homogeneous polynomials expressed in terms of binomials of the form $$x_i+y_j$$ for $i,j\in\mathbb N$ with integer coefficients that I conjecture can have nonnegative coefficients when ...
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Dimensions of singular loci of Schubert varieties

Let $X_w$ denote a Schubert variety (in type $A$). Lakshmibai and Sandhya proved that $X_w$ is smooth if and only if $w$ avoids the patterns $3412$ and $4231$; in general the singular locus of ...
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Cohomology ring of grassmannian and Pieri rule

I am learning Schubert variety and I came across a problem to understand a particular detail (I asked the same question on mathoverflow : https://mathoverflow.net/questions/397999/cohomology-ring-of-...
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reference for cohomology ring of even orthogonal grassmannian

Even orthogonal Grassmannian $OG(m,2n)$ are the spaces parameterize $m$-dimensianl isotropic subspaces in a vector space $V\simeq \mathbb{C}^{2n}$, with a nondegenerate symmetric bilinear form. It's ...
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Is the Kempf-Laksov-resolution of a Gorenstein single-condition Schubert variety a blowup?

Let $Gr(k,V)$ be the Grassmann bundle of $k$-dimensional subspaces of an $n$-dimensional vector space $V$ equipped with a full flag $0=E_0\subset E_1 \subset \ldots \subset E_{n-1}\subset E_n=V$. ...
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Reference request: cohomology ring of flag varieties

Just when I started understanding the basics of Schubert calculus and how the cohomology ring of Grassmannians $G(k,n)$ works, I figured I needed a generalization in terms of (partial) flags. My goal ...
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Degree of a subvariety of the Grassmannian

This might be a stupid question due to my lack of knowledge in algebraic geometry. So I have a subvariety $V\subset G_k(\mathbb{C}^n)$ of codimension $k$. The only thing I know about $V$ is its ...
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Schubert calculus

Let $X = Gr(2,4)$ the complex Grassmannian of $2$-planes in $V = \Bbb C^4$ and $S$ the tautological bundle, $Q$ the quotient bundle. The cohomology ring is generated by $c_1(S), c_2(S)$ with relations ...
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Schubert condition $\dim(X\cap E^{i+N-1})\geq i$ for $1\leq i\leq q$ with $X\subset E^{N+q}$ gives $qN-i$ cycle?

Let $X$ be a q-dimensional linear space through $0$ in $E^{q+N}$ where $q+N$ is the dimension of ambient vector space. Let $E^0\subset E^1\subset \dots\subset E^{q+N}$ be the filtration of ambient ...
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Faulty calculation in Schubert calculus: the number of lines on the intersection of two planes.

As a easy test of the correctness of the results of Schubert calculus I tried to solve the following problem with the symbolic method. Q: How many lines are there lying on the intersection of two ...
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Generically transversally intersecting Schubert cycles

I have a question about the the proof of Pieri's formula from Harris' and Eisenbuds's lecture "3264 and All That"on page 146. Before the proof we use this terminology (see page 139): let $G=G(k,V)$...
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Elementary description for $\mathbb{P}^1\times X$ and rational equivalence

In order to define the chow group of a variety $X$ one considers cycles modulo rational equivalence where two cycles $A_0, A_1\in Z(X)$ are called rationally equivalent if there exists a cycle on $\...
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Basis theorem for the Grassamannian [duplicate]

Does anyone know where I can find a good proof for the basis theorem of the cohomology ring of the Grassmannian, or give me a sketch of the proof? I'm already familiar with basic Schubert Calculus.
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Schubert cycles that intersect generically transversely.

Let $\mathcal{V}= 0 \subset V_1 \subset \cdots \subset V_{n-1}\subset V_n=V$, $\mathcal{W}=0 \subset W_1 \subset \cdots \subset W_{n-1} \subset W_n=W$ be two flags. We say that $\mathcal{V}$ and $\...
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closure of a Schubert cell

I'm trying to understand Schubert cells. I've just seen the definition and its connection with Young diagrams. What does the closure of a Schubert cell look like? I'm having trouble how to even think ...
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Questions on grassmannians from Harris' and Griffith's principles of algebraic geometry

I have a couple of questions about the explanations on the Grassmannian from Harris' and Griffith's "Principles of algebraic geometry" on page 194/195. We consider for $k \le n$ the $k$-th ...
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Schubert cell of intersection of subspaces

Assume you are given two subspace $V$ and $W$, belonging to some Schubert cells $C_I$ and $C_J$. Is there an elementary closed form description of $V\cap W$ in $C_{I\cap J}$? Here by "elementary" I ...
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What, exactly, is Schubert's symbolic calculus?

Most modern treatments of the Schubert calculus typically write about the cohomology ring of the Grassmannian. They also write, almost as an afterthought, that this is derived from Schubert's "...
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Reference for Grassmann and Schubert varieties for Beginners .

I need some references to understand Grassmann and Schubert Variety as a beginner. I am looking for self-contained notes on these. Thanks.
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Schubert Cell Structure for some variety with prescribed bilinear form

We know the construction of Schubert cell complex structure for the case of Grassmann manifold $Gr_k(\mathbb{C}^n)$ i.e variety of all k- dimensional subspaces $V$ of $\mathbb{C}^n$. Consider that ...
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Unipotent action on flag variety

I am reading https://www.sciencedirect.com/science/article/pii/S138572587680008X and must be missing something obvious. The premise is that we are considering the fixed point set of a unipotent ...
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$H^*(\mathbb{G}(k,n))$ free with basis the partitions that index the Schubert varieties [duplicate]

I have to proof the fact that $H^*(\mathbb{G}(k,n))$ is, as an abelian group, free with basis the partitions that index the Schubert varieties, but I'm having trouble doing it myself. Denote $\sigma_{...
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Every Schubert cycle a Chern class?

Consider the Grassmann variety $\mathbb{G}(k,n)$ and its Chow ring $A$. It is known that the classes of Schubert cycles form a $\mathbb{Z}$ basis of $A$. Is it known which of these Schubert cycles can ...
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Schubert class in the Grassmannian G(3,6)

How to compute the Schubert class $\sigma$$^2$$_2$$_1$ in the Grassmannian G(3,6)? I remember the result is $\sigma$$_3$$_3$ + 2$\sigma$$_3$$_2$$_1$ + $\sigma$$_2$$_2$$_2$.
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Singular Schubert Variety in $Fl_4(\mathbb C)$

Consider the Complete Flag Variety $Fl_4(\mathbb C)$ and its Schubert Variety given by the permutation $(1234) \to (3142)$, i.e. its highest dimension cell can be parametrized by $$\begin{bmatrix} ...
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How to prove Wielandt minimax formula?

The statements are as follows: Let $1\leqslant i_1<i_2<\cdots<i_k\leqslant n$ be integers. Define a partial flag to be a nested collection $V_1\subset V_2\cdots\subset V_k$ of subspaces of $\...
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Schubert variety associated to a flag of subspaces of a vector space

At the end of the following page : The associated Schubert variety of a flag of subspaces of a vector space. , the author says : Let $[W]\in X=X_{\underline i}$, by construction: \begin{equation} [W]=...
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Orthogonal Grassmannian

The orthogonal Grassmannian $OG(k,n)$ is the set of all isotropic $k$ dimensional subspaces of a $n$ dimensional subspaces with respect to a non-degenerate symmetric bilinear form. Is it isomorphic to ...
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Reference for flag varieties G/P

Is there a good reference for learning about flag varieties $G/P$? I'm already comfortable with the algebraic geometry and the example of Grassmannians, but I am not so comfortable with algebraic ...
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Understanding Schubert Varieties

I'm reading the paper "Chern Classes of Schubert Cells and Varieties" by Paolo Aluffi and Leonardo Constantin Mihalcea. I'm going through this paper rather slowly, but I'm a bit stuck on the second ...
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How can I get the matrix form for a schubert cell?

I am trying to learn how to understand the cell-decomposition for the grassmannian and am following these notes: http://www.math.drexel.edu/~jblasiak/grassmannian.pdf. On page 2 the author considers ...
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Is the quotient of standard parabolic subgroups isomorphic to a Schubert variety

Let $G$ be a reductive algebraic group over an algebraically closed field. Let $B \subseteq P_2 \subseteq P_1$ be a Borel subgroup and two parabolic subgroups. Then $P_1/P_2 \subseteq G/P_2$ is a ...
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$\mathbb P^1(\mathbb C)$ inside $Gr(2,4)$

Let $V$ be a vector space over $\mathbb C$ of dimension $4$ and consider the Grassmanian $Gr(2,V)$. Fix an index $I=(1,3)$ and a complete flag of $V=\langle e_1, e_2, e_3, e_4 \rangle$ given by $ F =\...
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Is the geometrical meaning of cup product still valid for subvarieties?

It is known that cup product is Poincaré dual to the intersection. I'm referring to the following fact: if $X$ is a closed, oriented smooth manifold and $A, B$ are transverse-intersecting oriented ...
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$GL_n(F)$ acts on the flag variety

I have the following 2-part question as a homework assignment... Let $F$ be a field and $n\in \mathbb{Z}_{\geq 1}$. A full flag in the vector space $F^n$ is a chain of subspaces \begin{align} \{...
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Intersection of Schubert cycles

I want to compute intersection of the Schubert cell $\sigma_{(3,0)}$ with all the cells $\sigma_{a_1, a_2}$ in the grassmanian $G(2,5)$. I am not sure I am doing correctly but I can't see my mistake. ...
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3 votes
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Why is the Complete Flag Variety an algebraic variety?

Let $V$ be a $\mathbb C $ - vector space of dimension $n$. Let's consider the set $Fl(n)$ of all the complete flags $F_{\bullet}$ $$F_1 \subset F_2 \cdots \subset F_n$$ where the $F_i$ are subspaces ...
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Understanding the cohomology ring of the Grassmannian

Some background first: I'm trying to understand the solution of some enumerative geometry problems, such as proving that a smooth cubic contains $27$ lines. I know that this becomes easier once one ...
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Betti numbers for the isotropic grassmannian

I want to know if there is some type of combinatorial formula for computing the Betti numbers of the isotropic grassmannian $IG(r,2n)$ for $r\leq n$. I'm thinking of this as the homogenous space $G/P$...
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What is a general linear subspace?

When it comes to defining a general plane with respect to a line in $\mathbb R^3$, I can think of this definition as: take any plane not containing the line. Reading Fulton's "Young Tableau" I can't ...
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Schubert and Grassmann varieties

Can anybody please suggest nice reference which will have lots of examples and counter examples for studying Grassmann varieties in particular Schubert variety?
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Locus of tangent lines to a smooth curve of degree $d$ and genus $g$

Suppose $C\subseteq\mathbb{P}^3$ is a smooth curve of degree $d$ and genus $g$ (let's say we are working over $\mathbb{C}$). Let $T(C)$ be the locus of tangent lines to $C$. In other words, $$ T(C) = \...
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W action of Schubert varieties

Let $X(w)$ be a Schubert variety in $G/B$, where $G$ is a semisimple algebraic group and $B$ is a Borel in $G$. Then for $v \in W$, what is $v X(w)$ ? Is it same as $X(vw)$ ?
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Spherical Schubert Variety

I am studying Schubert variety and I came across a problem understand a particular detail. Let $G$ be a reductive group, and $\mu\in X_{\bullet}(T)$ a coweight i.e. $\mu\in Hom(\mathbb{G}_m,T)$, ...
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7 votes
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Number of fixed points of torus action over partial flag variety

Consider $g\in U(n)$ and $t\in T$, where $T$ is the diagonal maximal torus in $U(n)$. Some common manifolds may be obtained as quotients of the $U(n)$ like the complex grassmannian, $Gr(k,n)=U(n)/U(...
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Finding cohomology group of open dense subset of Schubert variety

Let $Y=Gr_{m}(\mathbb{C}^n)$ be the Grassmannian of $m$-plane inside $\mathbb{C}^n$. Let $X$ and $X'$ be two Schubert varieties inside $Y$ such that $X'\subset X$ and $dim(X')<dim(X)$. Let $Z=X\...
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Singular Locus of a Schubert variety

I am trying to compute the singular locus of the schubert variety $X_w$ in $G_{2,7}$ where $w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and ...
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Smoothness of Schubert Variety

Consider the Schubert variety $X(s_3s_2s_1s_4s_3s_2)$ in $SL_5/P_2$, where $P_2$ is the maximal parabolic corresponding to the simple root $\alpha_2$. In one line notation this permutation can be ...
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Applying the divided difference operator

This question is about divided difference operators. How do I perform $\partial _2$ or $\partial_3$ on $x_1^2x_2$? $\partial_i$ is defined as $\frac{p-r_i.p}{x_i-x_{i+1}}$, where $r_i$ is the ...
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