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Let $V$ be a $2n$-dimensional vector space endowed with a nondegenerate skew-symmetric form $q:V \to V^\vee$. We define the isotropic Grassmannian to be $$ X:=G_q(k,V)=\left\{[W] \in \mathbb P \left( …

Let $X$ be an isotropic Grassmannian, $Pic(X)=\mathbb Z$ (for example $X$ is a projective space or a quadric hypersurface). Consider a global section $s \in \Gamma(X,L)$, where $L$ is the generator of …

Let $V$ be a $(2n+1)$-dimensional complex vector space endowed with a non-degenerate symmetric bilinear form $q:V\times V\to \mathbb C$. Fix the notation: $$ OG(n-1,n,V):=\{W_{n-1}\subset W_n\subset W …

Let $V$ be a $\mathbb C$-vector space of dimension $n+2$ with a symmetric bilinear non-degenerate map $q: V \times V \to \mathbb C$. We define $$ G(k+1,V):=\{\mathbb PW \subset \mathbb PV : \dim W=k+1 …

Let us fix the setup over $\mathbb C$. Let $V$ be a $n$-dimensional vector space endowed with a non-degenerate symmetric bilinear form $q: V \times V \to \mathbb C$. We have that $$ Q^{n-2}=OG(1,V)=\{ …

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 rank 2 …

Then one can define the orthogonal Grassmannians as $$ OG(k,V)=\{\mathbb P W \subset \mathbb PV: \dim W=k \text{ and }W \text{ is }q-\text{isotropic}\} \subset G(k,V) $$ which is naturally contained in …

Let $X=S_{10} \subset \mathbb P^{15}$ be the 10-dimensional spinor variety in its minimal embedding. Consider a $\mathbb P^4 \subset S_{10}$, hence we can define the normal bundle $N=N_{\mathbb P^4|S_ …

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Ext{Ext}$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 conventi …

Having defined the spinor bundles over the orthogonal Grassmannians, can we produce some short exact sequences like the ones in the theorem above? …

Finally the question: are there similar expressions of the tangent bundles for the other orthogonal and isotropic Grassmannians? Also a reference would be enough. …

Moreover, the homogeneous $G$-variety $G/P \cap Q$ dominates the generalized Grassmannians $G/P$ and $G/Q$. $\require{AMScd}$ \begin{CD} G/P \cap Q @>{q}>> G/Q\\ @V{p}VV\\ G/P. …

Let $V$ be a $n$-dimensional complex vector space. Using Grothendieck's notation, we define the Grassmannian $G(k,V)$ as the space of $k$-quotients of $V$ or, equivalently, as $$ G(k,V)=\{ \mathbb P W …

Let us work over $\mathbb C$, using the Grothendieck projectivization $\mathbb P():=Proj(Sym())$. Consider a $7$-dimensional vector space $V$ endowed with a symmetric non-degenerate bilinear form $q:V …