\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 $\mathbb{P}^{6}$

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 \\ \mathrm{b}_{ 6 } &= 1 \\ \mathrm{b}_{ 7 } &= 1 \end{align*}
Basic information
dimension
6
index
7
Euler characteristic
7
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 7 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{6})$
$\mathrm{PGL}_{ 7 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{6})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{6})$
48
Projective geometry
minimal embedding

$\mathbb{P}^{6}\hookrightarrow\mathbb{P}^{ 6 }$

degree
1
Hilbert series
1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, ...
Exceptional collections
  • Beilinson constructed a full exceptional sequence in 1978, see MR0509388.
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.