# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Projective space $\mathbb{P}^{7}$

There exist other realisations of this Grassmannian:
Basic information
dimension
7
index
8
Euler characteristic
8
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 7 } = 1$, $\mathrm{b}_{ 8 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{7})$
$\mathrm{PGL}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{7})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{7})$
63
Projective geometry
minimal embedding

$\mathbb{P}^{7}\hookrightarrow\mathbb{P}^{ 7 }$

degree
1
Hilbert series
1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, 31824, 50388, 77520, 116280, 170544, 245157, 346104, 480700, 657800, ...
Exceptional collections
• Beilinson constructed a full exceptional sequence in 1978, see MR0509388.
• Kapranov constructed a full exceptional sequence in 1988, see MR0939472.