\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.

Grassmannian $\Gr(4,6)$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 5 } &= 3 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 7 } &= 2 \\ \mathrm{b}_{ 8 } &= 1 \\ \mathrm{b}_{ 9 } &= 1 \end{align*}
Basic information
dimension
8
index
6
Euler characteristic
15
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 7 } = 2$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 9 } = 1$
$\mathrm{Aut}^0(\Gr(4,6))$
$\mathrm{PGL}_{ 6 }$
$\pi_0\mathrm{Aut}(\Gr(4,6))$
$1$
$\dim\mathrm{Aut}^0(\Gr(4,6))$
35
Projective geometry
minimal embedding

$\Gr(4,6)\hookrightarrow\mathbb{P}^{ 14 }$

degree
14
Hilbert series
1, 15, 105, 490, 1764, 5292, 13860, 32670, 70785, 143143, 273273, 496860, 866320, 1456560, 2372112, 3755844, 5799465, 8756055, 12954865, 18818646, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.