\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 $\mathrm{Q}^{3}$

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 \end{align*}
Basic information
dimension
3
index
3
Euler characteristic
4
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$
$\mathrm{Aut}^0(\mathrm{Q}^{3})$
$\mathrm{SO}_{ 5 }$
$\pi_0\mathrm{Aut}(\mathrm{Q}^{3})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{Q}^{3})$
10
Projective geometry
minimal embedding

$\mathrm{Q}^{3}\hookrightarrow\mathbb{P}^{ 4 }$

degree
2
Hilbert series
1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, ...
Exceptional collections
  • Fonarev constructed a full exceptional sequence in 2019, see arXiv:1911.08968.
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.
  • Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.