\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, spinor variety $\OGr_+(5,10)$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 5 } &= 2 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 7 } &= 2 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 9 } &= 1 \\ \mathrm{b}_{ 10 } &= 1 \\ \mathrm{b}_{ 11 } &= 1 \end{align*}
Basic information
dimension
10
index
8
Euler characteristic
16
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 7 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 9 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 11 } = 1$
$\mathrm{Aut}^0(\OGr_+(5,10))$
$\mathrm{PSO}_{ 10 }$
$\pi_0\mathrm{Aut}(\OGr_+(5,10))$
$1$
$\dim\mathrm{Aut}^0(\OGr_+(5,10))$
45
Projective geometry
minimal embedding

$\OGr_+(5,10)\hookrightarrow\mathbb{P}^{ 15 }$

degree
12
Hilbert series
1, 16, 126, 672, 2772, 9504, 28314, 75504, 184041, 416416, 884884, 1782144, 3426384, 6325632, 11267532, 19442016, 32605881, 53300016, 85131970, 133138720, ...
Exceptional collections
  • Kuznetsov constructed a full exceptional sequence in 2006, see MR2238172.
  • Kuznetsov constructed a full exceptional sequence in 2008, see MR2238172.