\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 $\OGr(6,13)$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 5 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 7 } &= 4 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 9 } &= 4 \\ \mathrm{b}_{ 10 } &= 5 \\ \mathrm{b}_{ 11 } &= 5 \\ \mathrm{b}_{ 12 } &= 5 \\ \mathrm{b}_{ 13 } &= 5 \\ \mathrm{b}_{ 14 } &= 4 \\ \mathrm{b}_{ 15 } &= 4 \\ \mathrm{b}_{ 16 } &= 4 \\ \mathrm{b}_{ 17 } &= 3 \\ \mathrm{b}_{ 18 } &= 2 \\ \mathrm{b}_{ 19 } &= 2 \\ \mathrm{b}_{ 20 } &= 1 \\ \mathrm{b}_{ 21 } &= 1 \\ \mathrm{b}_{ 22 } &= 1 \end{align*}
Basic information
dimension
21
index
12
Euler characteristic
64
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 7 } = 4$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 9 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 11 } = 5$, $\mathrm{b}_{ 12 } = 5$, $\mathrm{b}_{ 13 } = 5$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 15 } = 4$, $\mathrm{b}_{ 16 } = 4$, $\mathrm{b}_{ 17 } = 3$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 19 } = 2$, $\mathrm{b}_{ 20 } = 1$, $\mathrm{b}_{ 21 } = 1$, $\mathrm{b}_{ 22 } = 1$
$\mathrm{Aut}^0(\OGr(6,13))$
$\mathrm{PSO}_{ 14 }$
$\pi_0\mathrm{Aut}(\OGr(6,13))$
$1$
$\dim\mathrm{Aut}^0(\OGr(6,13))$
91
Projective geometry
minimal embedding

$\OGr(6,13)\hookrightarrow\mathbb{P}^{ 63 }$

degree
33592
Hilbert series
1, 64, 1716, 27456, 306735, 2617472, 18076916, 105174784, 530803988, 2375626240, 9591590944, 35415105024, 120909381996, 385159298304, 1153471856900, 3268289298560, 8809061000025, 22690832049600, 56082454213500, 133468540920000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\OGr(6,13))$. 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?