\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 $\OGr(2,12)$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 5 } &= 4 \\ \mathrm{b}_{ 6 } &= 4 \\ \mathrm{b}_{ 7 } &= 5 \\ \mathrm{b}_{ 8 } &= 5 \\ \mathrm{b}_{ 9 } &= 6 \\ \mathrm{b}_{ 10 } &= 6 \\ \mathrm{b}_{ 11 } &= 5 \\ \mathrm{b}_{ 12 } &= 5 \\ \mathrm{b}_{ 13 } &= 4 \\ \mathrm{b}_{ 14 } &= 4 \\ \mathrm{b}_{ 15 } &= 2 \\ \mathrm{b}_{ 16 } &= 2 \\ \mathrm{b}_{ 17 } &= 1 \\ \mathrm{b}_{ 18 } &= 1 \end{align*}
Basic information
dimension
17
index
9
Euler characteristic
60
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 4$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 7 } = 5$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 9 } = 6$, $\mathrm{b}_{ 10 } = 6$, $\mathrm{b}_{ 11 } = 5$, $\mathrm{b}_{ 12 } = 5$, $\mathrm{b}_{ 13 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 15 } = 2$, $\mathrm{b}_{ 16 } = 2$, $\mathrm{b}_{ 17 } = 1$, $\mathrm{b}_{ 18 } = 1$
$\mathrm{Aut}^0(\OGr(2,12))$
$\mathrm{PSO}_{ 12 }$
$\pi_0\mathrm{Aut}(\OGr(2,12))$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\OGr(2,12))$
66
Projective geometry
minimal embedding

$\OGr(2,12)\hookrightarrow\mathbb{P}^{ 65 }$

degree
19448
Hilbert series
1, 66, 1638, 23100, 222156, 1613898, 9447438, 46562373, 199377750, 759230472, 2617486872, 8284996512, 24347884704, 67041815400, 174263649912, 430295153574, 1014662410839, 2295243043170, 4999983023750, 10524366180900, ...
Exceptional collections
  • Kuznetsov–Smirnov constructed a full exceptional sequence in 2020, see arXiv:2001.04148.