\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 $\Gr(2,8)$

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 } &= 3 \\ \mathrm{b}_{ 7 } &= 4 \\ \mathrm{b}_{ 8 } &= 3 \\ \mathrm{b}_{ 9 } &= 3 \\ \mathrm{b}_{ 10 } &= 2 \\ \mathrm{b}_{ 11 } &= 2 \\ \mathrm{b}_{ 12 } &= 1 \\ \mathrm{b}_{ 13 } &= 1 \end{align*}
Basic information
dimension
12
index
8
Euler characteristic
28
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 7 } = 4$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 9 } = 3$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 11 } = 2$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 13 } = 1$
$\mathrm{Aut}^0(\Gr(2,8))$
$\mathrm{PGL}_{ 8 }$
$\pi_0\mathrm{Aut}(\Gr(2,8))$
$1$
$\dim\mathrm{Aut}^0(\Gr(2,8))$
63
Projective geometry
minimal embedding

$\Gr(2,8)\hookrightarrow\mathbb{P}^{ 27 }$

degree
132
Hilbert series
1, 28, 336, 2520, 13860, 60984, 226512, 736164, 2147145, 5725720, 14158144, 32821152, 71954064, 150233760, 300467520, 578399976, 1075994073, 1941008916, 3405278800, 5824819000, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.