\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.

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

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \end{align*}
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.