\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 $\Gr(5,7)$

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 } &= 3 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 9 } &= 2 \\ \mathrm{b}_{ 10 } &= 1 \\ \mathrm{b}_{ 11 } &= 1 \end{align*}
Basic information
dimension
10
index
7
Euler characteristic
21
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 7 } = 3$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 9 } = 2$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 11 } = 1$
$\mathrm{Aut}^0(\Gr(5,7))$
$\mathrm{PGL}_{ 7 }$
$\pi_0\mathrm{Aut}(\Gr(5,7))$
$1$
$\dim\mathrm{Aut}^0(\Gr(5,7))$
48
Projective geometry
minimal embedding

$\Gr(5,7)\hookrightarrow\mathbb{P}^{ 20 }$

degree
42
Hilbert series
1, 21, 196, 1176, 5292, 19404, 60984, 169884, 429429, 1002001, 2186184, 4504864, 8836464, 16604784, 30046752, 52581816, 89311761, 147685461, 238369516, 376372920, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.