\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 $\Gr(3,6)$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 3 \\ \mathrm{b}_{ 5 } &= 3 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 7 } &= 3 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 9 } &= 1 \\ \mathrm{b}_{ 10 } &= 1 \end{align*}
Basic information
dimension
9
index
6
Euler characteristic
20
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 7 } = 3$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 9 } = 1$, $\mathrm{b}_{ 10 } = 1$
$\mathrm{Aut}^0(\Gr(3,6))$
$\mathrm{PGL}_{ 6 }$
$\pi_0\mathrm{Aut}(\Gr(3,6))$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\Gr(3,6))$
35
Projective geometry
minimal embedding

$\Gr(3,6)\hookrightarrow\mathbb{P}^{ 19 }$

degree
42
Hilbert series
1, 20, 175, 980, 4116, 14112, 41580, 108900, 259545, 572572, 1184183, 2318680, 4331600, 7768320, 13441968, 22535064, 36729945, 58373700, 90684055, 138003404, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.