OGr(2,17)—grassmannian.info

Betti numbers

\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 8 } &= 3 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 12 } &= 4 \\ \mathrm{b}_{ 14 } &= 4 \\ \mathrm{b}_{ 16 } &= 5 \\ \mathrm{b}_{ 18 } &= 5 \\ \mathrm{b}_{ 20 } &= 6 \\ \mathrm{b}_{ 22 } &= 6 \\ \mathrm{b}_{ 24 } &= 7 \\ \mathrm{b}_{ 26 } &= 7 \\ \mathrm{b}_{ 28 } &= 7 \\ \mathrm{b}_{ 30 } &= 7 \\ \mathrm{b}_{ 32 } &= 6 \\ \mathrm{b}_{ 34 } &= 6 \\ \mathrm{b}_{ 36 } &= 5 \\ \mathrm{b}_{ 38 } &= 5 \\ \mathrm{b}_{ 40 } &= 4 \\ \mathrm{b}_{ 42 } &= 4 \\ \mathrm{b}_{ 44 } &= 3 \\ \mathrm{b}_{ 46 } &= 3 \\ \mathrm{b}_{ 48 } &= 2 \\ \mathrm{b}_{ 50 } &= 2 \\ \mathrm{b}_{ 52 } &= 1 \\ \mathrm{b}_{ 54 } &= 1 \end{align*}

Basic information

dimension: 27
index: 14
Euler characteristic: 112
Betti numbers: $\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 18 } = 5$, $\mathrm{b}_{ 20 } = 6$, $\mathrm{b}_{ 22 } = 6$, $\mathrm{b}_{ 24 } = 7$, $\mathrm{b}_{ 26 } = 7$, $\mathrm{b}_{ 28 } = 7$, $\mathrm{b}_{ 30 } = 7$, $\mathrm{b}_{ 32 } = 6$, $\mathrm{b}_{ 34 } = 6$, $\mathrm{b}_{ 36 } = 5$, $\mathrm{b}_{ 38 } = 5$, $\mathrm{b}_{ 40 } = 4$, $\mathrm{b}_{ 42 } = 4$, $\mathrm{b}_{ 44 } = 3$, $\mathrm{b}_{ 46 } = 3$, $\mathrm{b}_{ 48 } = 2$, $\mathrm{b}_{ 50 } = 2$, $\mathrm{b}_{ 52 } = 1$, $\mathrm{b}_{ 54 } = 1$
$\mathrm{Aut}^0(\OGr(2,17))$: $\mathrm{SO}_{ 17 }$
$\pi_0\mathrm{Aut}(\OGr(2,17))$: $1$
$\dim\mathrm{Aut}^0(\OGr(2,17))$: 136

Projective geometry

minimal embedding: $\OGr(2,17)\hookrightarrow\mathbb{P}^{ 135 }$
degree: 10697760
Hilbert series: 1, 136, 6783, 186200, 3371984, 44667840, 462561840, 3921101136, 28137531300, 175319459392, 967438226524, 4802858127984, 21727215340880, 90511667240000, 350280152218800, 1268697624499800, 4327925830324425, 13981205126208600, 42973678970184375, 126195462478940400, ...

Projective duality

The projective dual is a hypersurface.

codimension: 1
degree: 112
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 $\mathbf D^{\mathrm b}(\OGr(2,17))$:

Kuznetsov (2008)
Samokhin–van der Kallen (2024)

Quantum cohomology

small: generically semisimple [Chaput–Perrin]
Adjoint, non-coadjoint case.
big: generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]
Convergent near the large-radius limit.

Homological projective duality

Full minimal rectangular Lefschetz collection. [Kuznetsov, Theorem 7.1]

Block sizes $(8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8)$; 112 objects.

Symmetric powers of the dual tautological bundle and a spinor bundle.

No HP-dual model recorded.

$k$-Fanoness

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

References

Kuznetsov, Alexander. Exceptional collections for Grassmannians of isotropic lines. Proc. Lond. Math. Soc. (3) 97 (2008), 155–182. MR2434094.
Chaput, P. E.; Perrin, N. On the quantum cohomology of adjoint varieties. Proc. Lond. Math. Soc. (3) 103 (2011), 294–330. arXiv:0904.4824. MR2821244.
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 $\OGr(2,17)$