# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Generalised Grassmannian of type F4/P3

Basic information
dimension
20
index
7
Euler characteristic
96
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 5 } = 4$, $\mathrm{b}_{ 6 } = 5$, $\mathrm{b}_{ 7 } = 6$, $\mathrm{b}_{ 8 } = 7$, $\mathrm{b}_{ 9 } = 7$, $\mathrm{b}_{ 10 } = 8$, $\mathrm{b}_{ 11 } = 8$, $\mathrm{b}_{ 12 } = 8$, $\mathrm{b}_{ 13 } = 7$, $\mathrm{b}_{ 14 } = 7$, $\mathrm{b}_{ 15 } = 6$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 17 } = 4$, $\mathrm{b}_{ 18 } = 3$, $\mathrm{b}_{ 19 } = 2$, $\mathrm{b}_{ 20 } = 1$, $\mathrm{b}_{ 21 } = 1$
$\mathrm{Aut}^0(\mathrm{F}_{4}/\mathrm{P}_{3})$
$\mathrm{F}_4$
$\pi_0\mathrm{Aut}(\mathrm{F}_{4}/\mathrm{P}_{3})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{F}_{4}/\mathrm{P}_{3})$
52
Projective geometry
minimal embedding

$\mathrm{F}_{4}/\mathrm{P}_{3}\hookrightarrow\mathbb{P}^{ 272 }$

degree
116093952
Hilbert series
1, 273, 19448, 629356, 11955216, 154187280, 1480023468, 11251379568, 70800674945, 380987317425, 1797111814680, 7575286952688, 28975319929088, 101819502301440, 332053131427056, 1013469861438216, 2915458573847313, 7952338413532513, 20672714506379936, 51443521206236940, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{F}_{4}/\mathrm{P}_{3})$. Will you be the first to construct one? Let us know if you do!