\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.

\mathrm{G}_2$-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
3
Euler characteristic
6
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$, $\mathrm{b}_{ 6 } = 1$
$\mathrm{Aut}^0(\mathrm{G}_{2}/\mathrm{P}_{2})$
$\mathrm{G}_2$
$\pi_0\mathrm{Aut}(\mathrm{G}_{2}/\mathrm{P}_{2})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{G}_{2}/\mathrm{P}_{2})$
14
Projective geometry
minimal embedding

$\mathrm{G}_{2}/\mathrm{P}_{2}\hookrightarrow\mathbb{P}^{ 13 }$

degree
18
Hilbert series
1, 14, 77, 273, 748, 1729, 3542, 6630, 11571, 19096, 30107, 45695, 67158, 96019, 134044, 183260, 245973, 324786, 422617, 542717, ...
Exceptional collections
  • Kuznetsov constructed a full exceptional sequence in 2008, see MR2238172.