\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.

Quadric $\mathrm{Q}^{8}$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 5 } &= 2 \\ \mathrm{b}_{ 6 } &= 1 \\ \mathrm{b}_{ 7 } &= 1 \\ \mathrm{b}_{ 8 } &= 1 \\ \mathrm{b}_{ 9 } &= 1 \end{align*}
Basic information
dimension
8
index
8
Euler characteristic
10
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 2$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 7 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 9 } = 1$
$\mathrm{Aut}^0(\mathrm{Q}^{8})$
$\mathrm{PSO}_{ 10 }$
$\pi_0\mathrm{Aut}(\mathrm{Q}^{8})$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\mathrm{Q}^{8})$
45
Projective geometry
minimal embedding

$\mathrm{Q}^{8}\hookrightarrow\mathbb{P}^{ 9 }$

degree
2
Hilbert series
1, 10, 54, 210, 660, 1782, 4290, 9438, 19305, 37180, 68068, 119340, 201552, 329460, 523260, 810084, 1225785, 1817046, 2643850, 3782350, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.