# 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.

78 questions
Filter by
Sorted by
Tagged with
67 views

### 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 $c(E)=1+c_1(E)+c_2(E)$...
• 1,269
125 views

### 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 ...
• 56k
18 views

### 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 ...
• 127
159 views

### 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-...
1 vote
95 views

### 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 ...
• 125
1 vote
40 views

### 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 $0=E_0\subset E_1 \subset \ldots \subset E_{n-1}\subset E_n=V$. ...
• 135
29 views

### 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 ...
• 658
83 views

### 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 ...
• 658
1 vote
86 views

### 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 ...
• 713
1 vote
27 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 ...
• 8,201
164 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 ...
• 418
89 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 $G=G(k,V)$...
71 views

• 440
117 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 ...
• 907
1 vote
126 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
38 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 ...
• 824
96 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 "...
• 6,096
99 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.
• 2,241
96 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 ...
• 821
1 vote
193 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 ...
• 759
52 views

• 3,958
290 views

• 3,958
114 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 ...
731 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} \{...
• 971
347 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. ...
646 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 ...
• 3,958
621 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 ...
146 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 $G/P$...
259 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 ...
• 3,958
98 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?
337 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) = \...
• 9,652
46 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 $X(vw)$ ?
• 49
202 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. $\mu\in Hom(\mathbb{G}_m,T)$, ...
• 403
501 views

• 3,428
92 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 ...
• 177
1 vote
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 ...
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 ...