# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Projective space $\mathbb{P}^{1}$

Basic information
dimension
1
index
2
Euler characteristic
2
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{1})$
$\mathrm{PGL}_{ 2 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{1})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{1})$
3
Projective geometry
minimal embedding

$\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{ 1 }$

degree
1
Hilbert series
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...
Exceptional collections
• Beilinson constructed a full exceptional sequence in 1978, see MR0509388.
• Kapranov constructed a full exceptional sequence in 1988, see MR0939472.