\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 $\mathrm{Q}^{4}$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 5 } &= 1 \end{align*}
Basic information
dimension
4
index
4
Euler characteristic
6
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$
$\mathrm{Aut}^0(\mathrm{Q}^{4})$
$\mathrm{PSO}_{ 6 }$
$\pi_0\mathrm{Aut}(\mathrm{Q}^{4})$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\mathrm{Q}^{4})$
15
Projective geometry
minimal embedding

$\mathrm{Q}^{4}\hookrightarrow\mathbb{P}^{ 5 }$

degree
2
Hilbert series
1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.