\begin{equation} \DeclareMathOperator\Gr{Gr} \DeclareMathOperator\LGr{LGr} \DeclareMathOperator\OGr{OGr} \DeclareMathOperator\SGr{SGr} \DeclareMathOperator\Kzero{K_0} \DeclareMathOperator\index{i} \DeclareMathOperator\rk{rk} \end{equation}

Grassmannian.info

A periodic table of (generalised) Grassmannians.

Orthogonal Grassmannian $\OGr(2,11)$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 5 } &= 3 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 7 } &= 4 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 9 } &= 4 \\ \mathrm{b}_{ 10 } &= 4 \\ \mathrm{b}_{ 11 } &= 3 \\ \mathrm{b}_{ 12 } &= 3 \\ \mathrm{b}_{ 13 } &= 2 \\ \mathrm{b}_{ 14 } &= 2 \\ \mathrm{b}_{ 15 } &= 1 \\ \mathrm{b}_{ 16 } &= 1 \end{align*}
Basic information
dimension
15
index
8
Euler characteristic
40
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 7 } = 4$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 9 } = 4$, $\mathrm{b}_{ 10 } = 4$, $\mathrm{b}_{ 11 } = 3$, $\mathrm{b}_{ 12 } = 3$, $\mathrm{b}_{ 13 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 15 } = 1$, $\mathrm{b}_{ 16 } = 1$
$\mathrm{Aut}^0(\OGr(2,11))$
$\mathrm{SO}_{ 11 }$
$\pi_0\mathrm{Aut}(\OGr(2,11))$
$1$
$\dim\mathrm{Aut}^0(\OGr(2,11))$
55
Projective geometry
minimal embedding

$\OGr(2,11)\hookrightarrow\mathbb{P}^{ 54 }$

degree
5720
Hilbert series
1, 55, 1144, 13650, 112200, 703494, 3586440, 15520791, 58790875, 199377750, 615879264, 1756322360, 4673587776, 11705713800, 27793245600, 62928540246, 136548771645, 285166559909, 575248883000, 1124398096250, ...
Exceptional collections
  • Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.