\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 $\mathbb{P}^{4}$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 5 } &= 1 \end{align*}
Basic information
dimension
4
index
5
Euler characteristic
5
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{4})$
$\mathrm{PGL}_{ 5 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{4})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{4})$
24
Projective geometry
minimal embedding

$\mathbb{P}^{4}\hookrightarrow\mathbb{P}^{ 4 }$

degree
1
Hilbert series
1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, ...
Exceptional collections
  • Beilinson constructed a full exceptional sequence in 1978, see MR0509388.
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.