\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 $\Gr(3,7)$

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 } &= 3 \\ \mathrm{b}_{ 5 } &= 4 \\ \mathrm{b}_{ 6 } &= 4 \\ \mathrm{b}_{ 7 } &= 5 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 9 } &= 4 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 11 } &= 2 \\ \mathrm{b}_{ 12 } &= 1 \\ \mathrm{b}_{ 13 } &= 1 \end{align*}
Basic information
dimension
12
index
7
Euler characteristic
35
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 5 } = 4$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 7 } = 5$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 9 } = 4$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 11 } = 2$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 13 } = 1$
$\mathrm{Aut}^0(\Gr(3,7))$
$\mathrm{PGL}_{ 7 }$
$\pi_0\mathrm{Aut}(\Gr(3,7))$
$1$
$\dim\mathrm{Aut}^0(\Gr(3,7))$
48
Projective geometry
minimal embedding

$\Gr(3,7)\hookrightarrow\mathbb{P}^{ 34 }$

degree
462
Hilbert series
1, 35, 490, 4116, 24696, 116424, 457380, 1557270, 4723719, 13026013, 33157124, 78835120, 176729280, 376375104, 766192176, 1498581756, 2828205765, 5168991135, 9177226366, 15870391460, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.