# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Projective space $\mathbb{P}^{2}$

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

$\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{ 2 }$

degree
1
Hilbert series
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, ...
Exceptional collections
• Beilinson constructed a full exceptional sequence in 1978, see MR0509388.
• Kapranov constructed a full exceptional sequence in 1988, see MR0939472.