\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.

Generalised Grassmannian of type F4/P1

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 } &= 2 \\ \mathrm{b}_{ 7 } &= 2 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 9 } &= 2 \\ \mathrm{b}_{ 10 } &= 2 \\ \mathrm{b}_{ 11 } &= 2 \\ \mathrm{b}_{ 12 } &= 2 \\ \mathrm{b}_{ 13 } &= 1 \\ \mathrm{b}_{ 14 } &= 1 \\ \mathrm{b}_{ 15 } &= 1 \\ \mathrm{b}_{ 16 } &= 1 \end{align*}
Basic information
dimension
15
index
8
Euler characteristic
24
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 7 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 9 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 11 } = 2$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 13 } = 1$, $\mathrm{b}_{ 14 } = 1$, $\mathrm{b}_{ 15 } = 1$, $\mathrm{b}_{ 16 } = 1$
$\mathrm{Aut}^0(\mathrm{F}_{4}/\mathrm{P}_{1})$
$\mathrm{F}_4$
$\pi_0\mathrm{Aut}(\mathrm{F}_{4}/\mathrm{P}_{1})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{F}_{4}/\mathrm{P}_{1})$
52
Projective geometry
minimal embedding

$\mathrm{F}_{4}/\mathrm{P}_{1}\hookrightarrow\mathbb{P}^{ 51 }$

degree
4992
Hilbert series
1, 52, 1053, 12376, 100776, 627912, 3187041, 13748020, 51949755, 175847880, 542393670, 1544927904, 4107092288, 10278624864, 24388573014, 55188666312, 119696471453, 249869263644, 503865726155, 984563860280, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{F}_{4}/\mathrm{P}_{1})$. Will you be the first to construct one? Let us know if you do!