\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 $\Gr(3,5)$

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 } &= 2 \\ \mathrm{b}_{ 6 } &= 1 \\ \mathrm{b}_{ 7 } &= 1 \end{align*}
Basic information
dimension
6
index
5
Euler characteristic
10
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 2$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 7 } = 1$
$\mathrm{Aut}^0(\Gr(3,5))$
$\mathrm{PGL}_{ 5 }$
$\pi_0\mathrm{Aut}(\Gr(3,5))$
$1$
$\dim\mathrm{Aut}^0(\Gr(3,5))$
24
Projective geometry
minimal embedding

$\Gr(3,5)\hookrightarrow\mathbb{P}^{ 9 }$

degree
5
Hilbert series
1, 10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.