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.

Filter by
Sorted by
Tagged with
1
vote
0answers
20 views

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 ...
4
votes
1answer
145 views

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 ...
3
votes
0answers
71 views

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 $G=G(k,V)$...
0
votes
0answers
40 views

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 $\...
0
votes
0answers
18 views

Properties of a general element of the intersection of two Schubert Cycles

We have the following Lemma: Lemma Let $\Sigma_a(\mathcal{V}),\Sigma_b(\mathcal{W})$ be two Schubert cycle defined relative to transverse flags $\mathcal{V}$ and $\mathcal{W}$. If $\Lambda \in \...
0
votes
0answers
15 views

which mehod can easily design index of Grassmannian and its k and n or any function for this in Macaulay2 and how to convert poset to this index?

how to know k and n and its index of Grassmannian? which mehod can easily design index of Grassmannian and its k and n or any library or function for this in Macaulay2? is there any library or ...
0
votes
0answers
61 views

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 $\...
0
votes
1answer
53 views

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 ...
1
vote
1answer
82 views

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 ...
1
vote
0answers
36 views

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 ...
2
votes
1answer
75 views

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 "...
3
votes
0answers
78 views

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.
1
vote
0answers
86 views

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 ...
1
vote
1answer
60 views

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 ...
0
votes
0answers
47 views

$H^*(\mathbb{G}(k,n))$ 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_{...
1
vote
0answers
90 views

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 ...
2
votes
1answer
295 views

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$.
2
votes
0answers
98 views

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} ...
2
votes
1answer
159 views

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 $\...
0
votes
0answers
94 views

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]=...
1
vote
0answers
389 views

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 ...
6
votes
0answers
208 views

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 ...
2
votes
0answers
53 views

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 ...
1
vote
0answers
77 views

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 ...
3
votes
0answers
135 views

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 ...
2
votes
1answer
93 views

$\mathbb P^1(\mathbb C)$ 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 =\...
4
votes
1answer
87 views

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 ...
4
votes
1answer
367 views

$GL_n(F)$ 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} \{...
2
votes
1answer
273 views

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. ...
3
votes
1answer
400 views

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 ...
2
votes
1answer
527 views

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 ...
3
votes
0answers
127 views

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 $G/P$...
0
votes
1answer
202 views

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 ...
0
votes
1answer
77 views

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?
4
votes
1answer
207 views

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) = \...
0
votes
0answers
43 views

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 $X(vw)$ ?
9
votes
0answers
179 views

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. $\mu\in Hom(\mathbb{G}_m,T)$, ...
7
votes
1answer
296 views

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(...
2
votes
1answer
96 views

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\...
2
votes
0answers
81 views

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 ...
1
vote
0answers
130 views

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 ...
1
vote
1answer
102 views

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 ...
1
vote
1answer
153 views

The associated Schubert variety of a flag of subspaces of a vector space.

Let $V$ be a vector space and $W_1 \subsetneq W_2 \subsetneq ... \subsetneq W_\ell \subsetneq V $ a flag of subspaces. The associated Schubert variety is defined as : $ \Omega ( W_{ \bullet } ) = \{ \...
4
votes
2answers
521 views

Schubert decomposition of a Grassmannian

I'm going through Sheldon Katz's Enumerative Geometry and String Theory, and a few things regarding the Grassmannian $G(2,4)$ (lines in projective space) are bothering me: How can I compute the ...
0
votes
1answer
320 views

Schubert cell decomposition and full flags

I am looking for a self-contained basic theory of Schubert-cells through finding the decomposition of the full flag $Fl_3(\mathbb C^3)$.
2
votes
1answer
150 views

Schubert Cells of Flags

I have been reading on these notes Undergraduate Lectures on Flag Varieties and I need some explanations on two things: In page 3, how he modefied the matrices in the "Second Attempt" In the same ...
1
vote
3answers
55 views

Dimension of a linear space in $\mathbb{P}^{n}$ (Schubert Calculus)

I am reading about Schubert calculus and have come across this definition of a linear space: A linear space $L$ in $\mathbb{P}^{n}$ is defined as the set of points $P = (p(0), p(1), \ldots, p(n))$ of ...
3
votes
2answers
254 views

Schubert calculus on Grassmannians

Can anyone please suggest me some notes or books where I can read about Schubert calculus? I am studying Grassmannian varieties so I would like to understand how to use this tool, in particular with ...
4
votes
2answers
2k views

Integral homology of real Grassmannian $G(2,4)$

I would like to compute $\pi_1$ and the integral homology groups of the real Grassmannian $G(2,4)$. (This is a question on an old qualifying exam.) The hint for the computation of $\pi_1$ is to put a ...
2
votes
1answer
91 views

Two questions about Schubert calculus and Schur functions.

I am reading the file. I have a question on pae 28. How to prove that $[X_{\{2,4\}}] = S_{(1)} = x_1 + x_2 + \cdots$ and $S_{(1)}^4 = 2 S_{(2,2)} + S_{(3,1)} + S_{(2,1,1)}$? I tried to verify $S_{(1)}^...