# Questions tagged [grassmannians]

Grassmannians are algebraic varieties whose points corresponds to vector subspaces of a fixed dimension in a fixed vector space.

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### Differential of a specific morphism to a Grassmannian

This is a problem that's been bugging me for some time, and therefore I've decided to ask it here. Let $X$ be a smooth projective (irreducible) variety over an algebraically closed field of ...
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### Universal hyperplane section and nondegeneracy of general hyperplane section

I have a question about Exercise 18.11 In Harris' book Algebraic Geometry, on page 231: Give a proof of the nondegeneracy of the general hyperplane section of an projective irreducible nondegenerated ...
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### Fundamental domain for two Grassmannians

Let $\pi_1, \pi_2$ be two $k$-dimensional subspaces of $\mathbb R^n$. Using elements of the orthogonal group $O(n)$, how much can we simplify $\pi_1, \pi_2$? Certainly there always exists $A \in O(n)$ ...
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### Extensions for a short exact sequence on Grassmannians

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Ext{Ext}$Let us consider a $n$-dimensional complex vector space $V$ and denote by $G(k,n)$ the Grassmannian of $k$-planes in $V$. We use the ...
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### Tautological bundle and its dual

Let $X=\mathbb G(2,V)$ be the Grassmannian of $2$-planes in $V=\mathbb C^n$. We denote by $\mathcal S$ the tautological bundle on $X$. In a paper there is written that "since $\mathcal S$ is a ...
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### Tangent bundle for orthogonal and isotropic Grassmannians

We will work over $\mathbb C$. Let us consider a $n$-dimensional vector space $V$, then we define the $k$-th Grassmannian as $$\mathbb G(k,V):=\{W \subset V : \dim W=k\}.$$ Then consider a non-...
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### Embedding of co-oriented subspaces into positive Grassmannian

$\def\R{\mathbb{R}}$Let $P_1$, $P_2$, $P_3$ be three $m$-dimensional subspaces in $\R^n$. With a slight abuse of notation they will also denote the ortho-projectors on the respective subspaces. We ...
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The complex projective line $\mathbb{P}^1(\mathbb{C})$ can be identified with the sphere at infinity of hyperbolic $3$-space, modeled say by the Poincare open $3$-ball in $\mathbb{R}^3$ (the sphere at ...