# Questions tagged [grassmannian]

In mathematics, the Grassmannian $\mathbf{Gr}(r, V)$ is a space which parameterizes all linear subspaces of a vector space $V$ of given dimension $r$.

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### Exercise 5.4.1 in Smith's Invitation to Algebraic Geometry; lines that are tangent to conic are closed subvariety of Gr$(2,3)$ in $\mathbb{P}^2$

I am trying to solve Exercise 5.4.1 in Karen E. Smith's Invatation to Algebraic Geometry: Fix an irreducible conic $C$ in $\mathbb{P}^2$. Show that the set of lines in $\mathbb{P}^2$ that fail to ...
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### If all the vector spaces in a convergent sequence in the Grassmanian contains a vector, then the limit of the sequence contains the vector?

Let $G(k,n)$ denote the Grassmanian of $k$-dimensional vector subspaces in $\mathbb R^n$. Suppose $W_n$, $n \in \mathbb N$ is a sequence of subspaces in $G(k,n)$ converging to some $W$ in its topology....
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### How to determine Plucker coordinates of Infinite dimension Sato Grassmannian? [closed]

I wish to determine Plucker coordinates of Infinite dimension Sato Grassmannian. I've been reading the original paper by Sato and other related works, but couldn't understand few things Do Plucker ...
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### Map from Orthonormal Frame to Grassmannian is Quotient Map

Let $\text{Or}_k(\mathbb{R}^n)$ be set of orthonormal $k$-frames for $k\leq n$ (collection of set of orthonormal $k$ vectors in $\mathbb{R}^n$) and let $\text{Gr}_k(\mathbb{R}^n)$ be Grassmannian of ...
1 vote
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### How to see the second fundamental form is symmetric when defining it as the derivative of the Gauss map?

Let $X \subset \mathbb C^N$ be an $n$-dimensional complex submanifold. Voisin writes in  Let us introduce the second fundamental form $$\Phi: S^2 T_{X,x} \to \mathbb C^N / T_{X,x}$$ which can be ...
1 vote
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### Proving the Grassmannian is a manifold

The Grassmannian $\mathrm{Gr}(k, V)$ is the space of all $k$-dimensional subspaces of a (real or complex) vector space $V$. I am interested in the following strategy of proving $\mathrm{Gr}(k, V)$ is ...
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1 vote
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### Cantor-like set in $\mathbb{R}^2$ is purely 1-unrectifiable

I was just reading about rectifiability, and an example of purely 1-unrectifiable set can be given as follows. Consider the Cantor-like set $C_{1/2}$ obtained like this: [0,1] \to [0, 1/4] \cup [3/4,...
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### Why isn't the dimension of the space of curvature-like tensors equal to the dimension of the Grassmannian?

It's a well known fact that if $V$ is a vector space of dimension $n$, then the vector space consisting of all curvature-like tensors (see this text for a precise definition and a proof) has dimension ...
1 vote
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### What is the dimension of relative Grassmannian?

Let $E$ be a $r$ rank vector bundle on a smooth projective curve $C$ of genus $g$. Let $G$ denote the Grassmannian of $(r-k)$ rank locally free quotients of $E$. A paper by Mukai-Sakai seems to claim ...
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### Grassmannian $G(m,n)$ is locally Euclidean of dimension $m(n-m)$

A generalization of projective spaces are the Grassmann manifolds. Consider $G(m,n)$ as the set of all $m$-dimensional subspaces of $\mathbb{R}^n$. The topology on $G(m,n)$ should then arise ...
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### lines in projective space with non-algebraically closed field

I apologize in advance for this possibly trivial question. Let $k$ be a field that is not necessarily algebraically closed. We consider $\mathbb{P}^n_k$ and let $l$ be a line in it. There is a ...
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### Group/H-space structures for standard models of classifying spaces?

Let $G$ be a commutative topological group. May in his textbook, A concise course in algebraic topology, gives a model of the classifying space $BG$ so that it is a commutative topological group ...
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### Unique representation of $\mathbf{Gr}^+(p,n)$ the oriented real Grassmannian

For $\mathbf{Gr}(p,n)$ the $p$ dimensional subspace of $\mathbb{R}^n$, or equivalently $O(n)/O(p)\times O(n-p)$, a point has a unique projector representation $P = UU’$ where U is an $n \times p$ ...
1 vote
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### Realizing a neighborhood in $\mathbf{Gr}(k,n)$ as a neighborhood in $\mathbf{Gr}(1,n)$

Fix two positive integers $k,n$ with $1 < k < n$ and equip the Grassmannian $\mathbf{Gr}(k,n)$ with your favorite metric $\rho$. Now fix $\delta > 0$ and consider an open ball $B_{\delta}(x)$ ...
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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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### Inverse of Grassmann variables

Let $\theta$ be a Grassmann variable. I know that $1/\theta$ is not defined but $\frac{1}{1-\theta}=1+\theta$. My question is simple: is $\frac{1}{\bar{\theta}\theta}$ defined and if so how? My guess ...
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### Rigorously working with flat limits: lines meeting a curve by specialization

I am trying to get comfortable with flat limits. This question is motivated by Section 3.5.3 of Eisenbud and Harris's '3264 And All That' and Exercises 3.35 and 3.36. This section and the surrounding ...
I had been thinking of Grass$(p,r)$ as the set of $r$ dim subspaces of $k^p$ where $k$ is whatever ground field we take in the context. We then topologize this set. Now in one of his papers, Nitsure ...