orthogonal Grassmannian, spinor variety OGr+(7,14)

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: $\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 7 } = 4$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 9 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 11 } = 5$, $\mathrm{b}_{ 12 } = 5$, $\mathrm{b}_{ 13 } = 5$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 15 } = 4$, $\mathrm{b}_{ 16 } = 4$, $\mathrm{b}_{ 17 } = 3$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 19 } = 2$, $\mathrm{b}_{ 20 } = 1$, $\mathrm{b}_{ 21 } = 1$, $\mathrm{b}_{ 22 } = 1$
$\mathrm{Aut}^0(\OGr_+(7,14))$: $\mathrm{PSO}_{ 14 }$
$\pi_0\mathrm{Aut}(\OGr_+(7,14))$: $1$
$\dim\mathrm{Aut}^0(\OGr_+(7,14))$: 91

Projective geometry

minimal embedding: $\OGr_+(7,14)\hookrightarrow\mathbb{P}^{ 63 }$
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 $\mathbf{D}^{\mathrm{b}}(\OGr_+(7,14))$. 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?

Grassmannian.info

Orthogonal Grassmannian, spinor variety $\OGr_+(7,14)$