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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Does the Grassmannian scheme have meet and join?
The case of a field
Let $\mathbf{k}$ be a field (for now). Let $n\in\mathbb{N}$, and set $\left[
n\right] :=\left\{ 1,2,\ldots,n\right\} $. Consider the $\mathbf{k}$-vector
space $V:=\mathbf{k}^{n}$...
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Reference for Clifford algebras and orthogonal grassmanians
I'm reading about spinor bundles in Kuznetsov's paper https://arxiv.org/pdf/math/0512013. On page 17 he states that the orthogonal Grassmanian $\mathsf{OGr}(m,2m)$ with respect to a non-degenerate ...
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Different results for a computation on cohomology of Grassmannian
I try to study a specific operation of pullback and pushforward related to flag varieties.
Let $Y=Gr(r-1,2r-1)$ be the variety of $r-1$ subspaces in a vector space of dimension $2r-1$. I also have $Y'=...
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Jacobian of the exponential map on the Grassmann manifold
Formulas for the Riemannian exponential map $\exp_{P}$ and its differential at a given tangent vector $\Delta\in T_PGr(n,p)$ $(d\exp_P)_{\Delta}$ can be computed for the Grassmann manifold $Gr(n,p)$ (...
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Morse decomposition of the Orthogonal groups — in the literature?
$\newcommand\O{O}%In case \\\$\operatorname O\\\$ might be acceptable, just change this to `\DeclareMathOperator\O{O}`
\DeclareMathOperator\tr{tr}$Let $\O_n$ be the orthogonal group of $n \times n$ ...
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Question about Kapranov's resolution of the diagonal for Grassmannians
I have been thinking about derived categories of homogeneous spaces a little bit lately, especially $\mathrm{D}^{\mathrm{b}}(\mathrm{Gr}(k,V))$. Some foundational results are proven in a paper of ...
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Two-point boundary problem for Jacobi fields on the Grassmann manifold
Jacobi fields on a Riemannian manifold can be expressed using the differential of the exponential map, given an initial value of the field $J(0)$ and its derivative $D_t J(0)$.
Is it also possible to ...
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counting integral points on grassmannians by modular forms
An integral point $P$ of a Grassmannian $Gr(k,n)$ is a $k$-dimensional subspace such that $P \cap \mathbb{Z}^n$ is a rank $k$ sublattice of $\mathbb{Z}^n$. Its height $H$ is given by the determinant ...
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Question regarding Hilbert scheme's construction
I have been recently studying the construction of the Hilbert scheme from the book 'Deformations of algebraic schemes' by Sernesi and I am stuck with one of the very first steps. Let me expose the ...
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Forms of Grassmannians
The question is complementary to this question. How to construct pointless real forms of Grassmannians $\operatorname{Gr}(k,n)$? For which $k$ and $n$ do they exist? Any reference will be very helpful....
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Map from the set of bases of lattices in finite characteristic to Grassmannian
Let $q$ be a power of a prime $p$, $\Bbb F_q[\theta]$ the analog of $\Bbb Z$ in characteristic $p$, and $\Bbb C_\infty$ the analog of $\Bbb C$ in characteristic $p$. A lattice of rank $r$ in $\Bbb C_\...
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Reference for quadratic presentation of the type C cluster algebra?
Related to this question but much simpler. I am looking for a reference to a description of the cluster algebra of finite type C in terms of generators and relations.
Specifically, I am interested in ...
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A strange compactification of the representation space of type A quivers
In my research, I came across the following strange family of varieties by trying to construct smooth varieties with a given Poincaré polynomial. Given $n\in \mathbb{N}$ and a tuple $(a_1, \ldots, a_n)...
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Coxeter group action on type C cluster algebra?
A basic question about finite type cluster algebras. Let $A$ denote the cluster algebra of type $\mathrm C_{n-1}$, $n\ge 3$. I will view $A$ as a $\mathbb Z$-algebra with the $n^2$ generators $\Delta_{...
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Intersecting a smooth hypersurface with planes
Let $X\subseteq \mathbb{P}_{\mathbb{C}}^n$ be a smooth hypersurface of degree $d\ge 4$, and consider the Grassmanian $G=\operatorname{Gr}(3,n+1)$ parametrizing the ($2$-)planes inside $\mathbb{P}_{\...
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