\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 $\mathrm{Q}^{5}$

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

$\mathrm{Q}^{5}\hookrightarrow\mathbb{P}^{ 6 }$

degree
2
Hilbert series
1, 7, 27, 77, 182, 378, 714, 1254, 2079, 3289, 5005, 7371, 10556, 14756, 20196, 27132, 35853, 46683, 59983, 76153, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.