\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.

Symplectic Grassmannian $\SGr(2,10)$

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 } &= 4 \\ \mathrm{b}_{ 9 } &= 4 \\ \mathrm{b}_{ 10 } &= 4 \\ \mathrm{b}_{ 11 } &= 3 \\ \mathrm{b}_{ 12 } &= 3 \\ \mathrm{b}_{ 13 } &= 2 \\ \mathrm{b}_{ 14 } &= 2 \\ \mathrm{b}_{ 15 } &= 1 \\ \mathrm{b}_{ 16 } &= 1 \end{align*}
Basic information
dimension
15
index
9
Euler characteristic
40
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 7 } = 4$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 9 } = 4$, $\mathrm{b}_{ 10 } = 4$, $\mathrm{b}_{ 11 } = 3$, $\mathrm{b}_{ 12 } = 3$, $\mathrm{b}_{ 13 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 15 } = 1$, $\mathrm{b}_{ 16 } = 1$
$\mathrm{Aut}^0(\SGr(2,10))$
$\mathrm{PSp}_{ 10 }$
$\pi_0\mathrm{Aut}(\SGr(2,10))$
$1$
$\dim\mathrm{Aut}^0(\SGr(2,10))$
55
Projective geometry
minimal embedding

$\SGr(2,10)\hookrightarrow\mathbb{P}^{ 43 }$

degree
1430
Hilbert series
1, 44, 780, 8250, 61710, 358644, 1717716, 7054905, 25561250, 83431920, 249284464, 690416376, 1790285640, 4381818000, 10190856720, 22647113346, 48317257659, 99361170700, 197627787500, 381317615250, ...
Exceptional collections
  • Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.