\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 $\OGr(2,7)$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 5 } &= 2 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 7 } &= 1 \\ \mathrm{b}_{ 8 } &= 1 \end{align*}
Basic information
dimension
7
index
4
Euler characteristic
12
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 7 } = 1$, $\mathrm{b}_{ 8 } = 1$
$\mathrm{Aut}^0(\OGr(2,7))$
$\mathrm{SO}_{ 7 }$
$\pi_0\mathrm{Aut}(\OGr(2,7))$
$1$
$\dim\mathrm{Aut}^0(\OGr(2,7))$
21
Projective geometry
minimal embedding

$\OGr(2,7)\hookrightarrow\mathbb{P}^{ 20 }$

degree
56
Hilbert series
1, 21, 168, 825, 3003, 8918, 22848, 52326, 109725, 214291, 394680, 692055, 1163799, 1887900, 2968064, 4539612, 6776217, 9897537, 14177800, 19955397, ...
Exceptional collections
  • Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.