\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.

Orthogonal Grassmannian $\OGr(3,9)$

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 } &= 4 \\ \mathrm{b}_{ 7 } &= 4 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 9 } &= 3 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 11 } &= 2 \\ \mathrm{b}_{ 12 } &= 1 \\ \mathrm{b}_{ 13 } &= 1 \end{align*}
Basic information
dimension
12
index
5
Euler characteristic
32
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 7 } = 4$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 9 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 11 } = 2$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 13 } = 1$
$\mathrm{Aut}^0(\OGr(3,9))$
$\mathrm{SO}_{ 9 }$
$\pi_0\mathrm{Aut}(\OGr(3,9))$
$1$
$\dim\mathrm{Aut}^0(\OGr(3,9))$
36
Projective geometry
minimal embedding

$\OGr(3,9)\hookrightarrow\mathbb{P}^{ 83 }$

degree
16896
Hilbert series
1, 84, 1980, 23595, 182182, 1039584, 4744224, 18226890, 61072935, 183065740, 500256900, 1264384485, 2989300860, 6670723840, 14153213184, 28722125124, 56029956957, 105507744420, 192465052780, 341148914799, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\OGr(3,9))$. Will you be the first to construct one? Let us know if you do!

Kuznetsov–Polishchuk have constructed an exceptional collection of maximal length in MR3463417. Can you prove it's full?