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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Proof of the Euler characteristic of the real Grassmannian $\mathbf{G}(k, n)$
I'm interested in proving the following statement,
Let $\textbf{G}$(k,n) the real grassmannian and $\chi_{n,k} := \chi(\textbf{G}(k,n))$, where $\chi$ is the Euler characteristic, then
$$\chi_{k,n} = \...
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The Grassmannian $G(k,n) \subset G(k,n+1)$ as the zero locus of the tautological sub-bundle
Following my question
Pushforward of structure sheaf with respect to canonical inclusion of Grassmannian
I have a follow up question I would like to ask. I would have put this as a comment, but I feel ...
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Pushforward of structure sheaf with respect to canonical inclusion of Grassmannian
Let $j: Gr^{k,n} \rightarrow Gr^{k,n+1}$ be the canonical inclusion of Grassmannians (working over a field for simplicity). I am interested the pushforward $j_{*}\mathcal{O}_{Gr^{k,n}}$. Specifically, ...
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Minimum subset of the Grassmanian that covers all of a vector space $\mathbb{F}_d^n$
Consider a finite field $\mathbb{F}_d$ of order $d$, and let the vector space $V=\mathbb{F}_d^n$. Let $\mathbf{Gr}(m,V)$ be the Grassmanian containing all subspaces in $V$ of dimension $m$. Suppose $S\...
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Dimension of the Hilbert Scheme of conics
I am trying to compute the dimension of the Hilbert Scheme of conics in $\mathbb{P}^4$ $Hilb_{2T+1}(\mathbb{P}^4)$. I started with conics lying on a plane, so, taking the ideal $I=(Q,H_1,H_2)$ for two ...
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Geometric fixed-points of MO
Recall that the value of the orthogonal spectrum $\mathbf{MO}$ at an inner product space $V$ is the Thom space of the tautological bundle over the Grassmannian of $|V|$-demensiomal planes in $V\oplus ...
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Details on: Flag incidence variety is a projective variety
Let $0 \leq p \leq q \leq n$. Define the flag incidence variety
$$\text{Fl}(p,q,n):=\{(V,W) : V \leq W, V \in \text{G}(p,n), W \in \text{G}(q,n)\}$$
where $\mathrm{G}(i,n)$ is the Grassmannian, i.e. ...
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Generic planar sections of a projective variety
In an article by Broberg and Salberger, it is stated that
The set of pairs $(\Lambda,F)\in\mathbb G(k,n)\times \mathbb P\left(\mathbb Q_d[X_0,\ldots,X_n]\right)$ for which $\Lambda\cap V(F)$ is ...
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About the projection of a variety from a generic plane being birational
Take $Z\subseteq \mathbb P^N$ an irreducible subvariety of dimension $m$. For $\Lambda$ a projective subspace of dimension $N-m-2$ not intersecting $Z$, define $$\rho_\Lambda:Z\rightarrow\mathbb G(N-m-...
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Why the tautological bundle of the Grassmannian has only zero as global sections?
I'm in a course on Complex Geometry, and I have been studying the Grassmannian $G_r(\mathbb{C}^N)$ as a complex manifold and the construction of its tautological bundle. As in the case of the ...
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The universal bundle $\gamma_k\rightarrow BO_k$ is a real vector bundle
For all $n$, let $\gamma_k^n$ bet the tautological bundle over $Gr_k(\mathbb R^n)$, i.e.
$$\gamma_k^n=\{(V,v):V\in Gr_k(\mathbb R^n), v\in V\}$$
This is also naturally identified with the associated ...
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What is the action of $O(k)$ on $V_k(\mathbb R^n)$ making it a principal bundle?
Let $V_k(\mathbb R^n)$ be the Stiefel manifold of ordered $k$-tuples of vectors in $\mathbb R^n$. I have seen in many places that $V_k(\mathbb R^n)$ is an $O(k)$ principal bundle over the Grassmanian ...
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Blowup of diagonal in $\mathbb{P}^r \times \mathbb{P}^r$
Let $X= \mathbb{P}^r \times \mathbb{P}^r$. Suppose we blowup $X$ along $\Delta$ the diagonal to get $\tilde X$. I want to show that this is isomorphic to the fibre product which I describe below -
Let ...
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Can we construct the exterior algebra just from simple multivectors?
$
\newcommand\K{\mathbb K}
\newcommand\Ext{\mathop{\textstyle\bigwedge}}
\newcommand\Lip{\mathrm{Lip}}
\newcommand\ev{\mathrm{ev}}
\newcommand\Gr{\mathrm{Gr}}
$Let $V$ be a finite-dimensional $\K$-...
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Induced morphism of smooth variety to the Grassmannian on global sections?
In Huybrechts & Lehn, The Geometry of Moduli Spaces of Sheaves, page 143, it reads
Let $X$ be a smooth variety. Suppose $E$ is a locally free sheaf of rank $r$ which is generated by its space of ...
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