# 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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### Reference request: Détailed explanation why the Grassmannian scheme represents the Grassmannian functor

Similar questions have been asked on this site, including by myself, but none of these have been given a satisfying answer. The question is: Why does the Grassmannian scheme represent the Grassmannian ...
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### Definition of tautological vector bundle [closed]

Could you please give a detailed definition (or construction)of tautological vector bundle of Grassmannian over arbitrary base scheme? Thank you in advance!
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### Jumping conics in Grassmannians

Let $Gr(1,n)$ be the Grassmannian of lines in $\mathbb{P}^n$, and $f:\mathbb{P}^1\rightarrow Gr(1,n)$ a morphism of degree two. The pull-back $f^{*}S$ of the tautological bundle $S$ on $Gr(1,n)$ ...
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### 3-secant lines of a projective curve

Consider a smooth projective curve $C\subset\mathbb{P}^n$. Let $G(1,n)$ the Grassmannian of lines of $\mathbb{P}^n$. The variety $S_2(C)\subset G(1,n)$ parametrizing lines that are secant to $C$ (i.e.,...
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### number of maxima, minima, and saddle points on a Grassmannian

I have a positive, smooth function on a Grassmannian. I am wondering whether there exists any relation between the numbers of maxima, minima, and saddle points. At least on a 2-sphere, we have the ...
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### Sheaf cohomology of Grassmannian G(2,4) with values in twisted tautological bundles over an arbitrary field

Let k be an arbitrary field. Let $G(2,4)_k$ be the Grassmannian of 2-planes in 4-space over that field. Let $\mathcal{E}$ be the tautological quotient bundle on the Grassmannian. I am trying to ...
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### Grassmannian cluster algebra of infinite type has no trees in its mutation class

The question is why the statement in the title is true (is it?). To elaborate, recall that Grassmannian cluster algebra, according to Scott`s paper Grassmannians and Cluster Algebras, is the cluster ...
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### When can one continuously prescribe a unit vector orthogonal to a given orthonormal system?

Let $1 \leq k < n$ be natural numbers. Given orthonormal vectors $u_1,\dots,u_k$ in ${\bf R}^n$, one can always find an additional unit vector $v \in {\bf R}^n$ that is orthogonal to the preceding ...
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### another extremal property of regular polygons

$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}\newcommand{\E}{\mathbb{1}}$ In 1984 S.D.Berman, a Soviet mathematician, ...
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