OGr(5,14)—grassmannian.info

Betti numbers

\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 3 \\ \mathrm{b}_{ 6 } &= 4 \\ \mathrm{b}_{ 8 } &= 7 \\ \mathrm{b}_{ 10 } &= 10 \\ \mathrm{b}_{ 12 } &= 14 \\ \mathrm{b}_{ 14 } &= 18 \\ \mathrm{b}_{ 16 } &= 23 \\ \mathrm{b}_{ 18 } &= 28 \\ \mathrm{b}_{ 20 } &= 33 \\ \mathrm{b}_{ 22 } &= 38 \\ \mathrm{b}_{ 24 } &= 41 \\ \mathrm{b}_{ 26 } &= 45 \\ \mathrm{b}_{ 28 } &= 46 \\ \mathrm{b}_{ 30 } &= 48 \\ \mathrm{b}_{ 32 } &= 46 \\ \mathrm{b}_{ 34 } &= 45 \\ \mathrm{b}_{ 36 } &= 41 \\ \mathrm{b}_{ 38 } &= 38 \\ \mathrm{b}_{ 40 } &= 33 \\ \mathrm{b}_{ 42 } &= 28 \\ \mathrm{b}_{ 44 } &= 23 \\ \mathrm{b}_{ 46 } &= 18 \\ \mathrm{b}_{ 48 } &= 14 \\ \mathrm{b}_{ 50 } &= 10 \\ \mathrm{b}_{ 52 } &= 7 \\ \mathrm{b}_{ 54 } &= 4 \\ \mathrm{b}_{ 56 } &= 3 \\ \mathrm{b}_{ 58 } &= 1 \\ \mathrm{b}_{ 60 } &= 1 \end{align*}

Basic information

dimension: 30
index: 8
Euler characteristic: 672
Betti numbers: $\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 8 } = 7$, $\mathrm{b}_{ 10 } = 10$, $\mathrm{b}_{ 12 } = 14$, $\mathrm{b}_{ 14 } = 18$, $\mathrm{b}_{ 16 } = 23$, $\mathrm{b}_{ 18 } = 28$, $\mathrm{b}_{ 20 } = 33$, $\mathrm{b}_{ 22 } = 38$, $\mathrm{b}_{ 24 } = 41$, $\mathrm{b}_{ 26 } = 45$, $\mathrm{b}_{ 28 } = 46$, $\mathrm{b}_{ 30 } = 48$, $\mathrm{b}_{ 32 } = 46$, $\mathrm{b}_{ 34 } = 45$, $\mathrm{b}_{ 36 } = 41$, $\mathrm{b}_{ 38 } = 38$, $\mathrm{b}_{ 40 } = 33$, $\mathrm{b}_{ 42 } = 28$, $\mathrm{b}_{ 44 } = 23$, $\mathrm{b}_{ 46 } = 18$, $\mathrm{b}_{ 48 } = 14$, $\mathrm{b}_{ 50 } = 10$, $\mathrm{b}_{ 52 } = 7$, $\mathrm{b}_{ 54 } = 4$, $\mathrm{b}_{ 56 } = 3$, $\mathrm{b}_{ 58 } = 1$, $\mathrm{b}_{ 60 } = 1$
$\mathrm{Aut}^0(\OGr(5,14))$: $\mathrm{PSO}_{ 14 }$
$\pi_0\mathrm{Aut}(\OGr(5,14))$: $\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\OGr(5,14))$: 91

Projective geometry

minimal embedding: $\OGr(5,14)\hookrightarrow\mathbb{P}^{ 2001 }$
degree: 2066989350666240
Hilbert series: 1, 2002, 728728, 94602144, 6044141376, 229046390496, 5802989749128, 106690072541910, 1509741098290425, 17179475781768300, 162558611226123840, 1313147453795294208, 9247724296113278976, 57752947639463003136, 324357843962307939264, 1657540599815747369268, 7783448656263298835817, 33867812297575876585966, 137540596658540582944360, 524567465971843017724000, ...

Projective duality

The projective dual is a hypersurface.

codimension: 1
degree: 20793259313248
self-dual: no

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] [Tevelev, Theorems 7.1–7.2 (arXiv version)] All Grassmannians.

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\OGr(5,14))$:

Samokhin–van der Kallen (2024)

Quantum cohomology

small: semisimplicity unknown
big: generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]
Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\OGr(5,14))$:

Homological projective duality

No Lefschetz decomposition recorded.

No HP-dual model recorded.

$k$-Fanoness

Fano but not 2-Fano. [Araujo et al., Theorems 1.3 and 1.4]

References

Carolina Araujo; Roya Beheshti; Ana-Maria Castravet; Kelly Jabbusch; Svetlana Makarova; Enrica Mazzon; Libby Taylor; Nivedita Viswanathan. Higher Fano Manifolds. Rev. Un. Mat. Argentina 64 (2022), 103–125. arXiv:2110.02339.
Alexander Samokhin; Wilberd van der Kallen. Highest weight category structures on $\operatorname{rep}(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$. Preprint (2024). arXiv:2407.13653.
Changzheng Li; Zhaoyang Liu; Tony Yue Yu; Chi Zhang. Convergence and semisimplicity of big quantum cohomology of flag varieties. Preprint (2026). arXiv:2610.28418.

All references.

Grassmannian.info

Orthogonal Grassmannian $\OGr(5,14)$