\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, spinor variety $\OGr_+(10,20)$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 12 } &= 4 \\ \mathrm{b}_{ 14 } &= 5 \\ \mathrm{b}_{ 16 } &= 6 \\ \mathrm{b}_{ 18 } &= 8 \\ \mathrm{b}_{ 20 } &= 9 \\ \mathrm{b}_{ 22 } &= 10 \\ \mathrm{b}_{ 24 } &= 12 \\ \mathrm{b}_{ 26 } &= 13 \\ \mathrm{b}_{ 28 } &= 15 \\ \mathrm{b}_{ 30 } &= 17 \\ \mathrm{b}_{ 32 } &= 18 \\ \mathrm{b}_{ 34 } &= 19 \\ \mathrm{b}_{ 36 } &= 21 \\ \mathrm{b}_{ 38 } &= 21 \\ \mathrm{b}_{ 40 } &= 22 \\ \mathrm{b}_{ 42 } &= 23 \\ \mathrm{b}_{ 44 } &= 23 \\ \mathrm{b}_{ 46 } &= 23 \\ \mathrm{b}_{ 48 } &= 23 \\ \mathrm{b}_{ 50 } &= 22 \\ \mathrm{b}_{ 52 } &= 21 \\ \mathrm{b}_{ 54 } &= 21 \\ \mathrm{b}_{ 56 } &= 19 \\ \mathrm{b}_{ 58 } &= 18 \\ \mathrm{b}_{ 60 } &= 17 \\ \mathrm{b}_{ 62 } &= 15 \\ \mathrm{b}_{ 64 } &= 13 \\ \mathrm{b}_{ 66 } &= 12 \\ \mathrm{b}_{ 68 } &= 10 \\ \mathrm{b}_{ 70 } &= 9 \\ \mathrm{b}_{ 72 } &= 8 \\ \mathrm{b}_{ 74 } &= 6 \\ \mathrm{b}_{ 76 } &= 5 \\ \mathrm{b}_{ 78 } &= 4 \\ \mathrm{b}_{ 80 } &= 3 \\ \mathrm{b}_{ 82 } &= 2 \\ \mathrm{b}_{ 84 } &= 2 \\ \mathrm{b}_{ 86 } &= 1 \\ \mathrm{b}_{ 88 } &= 1 \\ \mathrm{b}_{ 90 } &= 1 \end{align*}
Basic information
dimension
45
index
18
Euler characteristic
512
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 5$, $\mathrm{b}_{ 16 } = 6$, $\mathrm{b}_{ 18 } = 8$, $\mathrm{b}_{ 20 } = 9$, $\mathrm{b}_{ 22 } = 10$, $\mathrm{b}_{ 24 } = 12$, $\mathrm{b}_{ 26 } = 13$, $\mathrm{b}_{ 28 } = 15$, $\mathrm{b}_{ 30 } = 17$, $\mathrm{b}_{ 32 } = 18$, $\mathrm{b}_{ 34 } = 19$, $\mathrm{b}_{ 36 } = 21$, $\mathrm{b}_{ 38 } = 21$, $\mathrm{b}_{ 40 } = 22$, $\mathrm{b}_{ 42 } = 23$, $\mathrm{b}_{ 44 } = 23$, $\mathrm{b}_{ 46 } = 23$, $\mathrm{b}_{ 48 } = 23$, $\mathrm{b}_{ 50 } = 22$, $\mathrm{b}_{ 52 } = 21$, $\mathrm{b}_{ 54 } = 21$, $\mathrm{b}_{ 56 } = 19$, $\mathrm{b}_{ 58 } = 18$, $\mathrm{b}_{ 60 } = 17$, $\mathrm{b}_{ 62 } = 15$, $\mathrm{b}_{ 64 } = 13$, $\mathrm{b}_{ 66 } = 12$, $\mathrm{b}_{ 68 } = 10$, $\mathrm{b}_{ 70 } = 9$, $\mathrm{b}_{ 72 } = 8$, $\mathrm{b}_{ 74 } = 6$, $\mathrm{b}_{ 76 } = 5$, $\mathrm{b}_{ 78 } = 4$, $\mathrm{b}_{ 80 } = 3$, $\mathrm{b}_{ 82 } = 2$, $\mathrm{b}_{ 84 } = 2$, $\mathrm{b}_{ 86 } = 1$, $\mathrm{b}_{ 88 } = 1$, $\mathrm{b}_{ 90 } = 1$
$\mathrm{Aut}^0(\OGr_+(10,20))$
$\mathrm{PSO}_{ 20 }$
$\pi_0\mathrm{Aut}(\OGr_+(10,20))$
$1$
$\dim\mathrm{Aut}^0(\OGr_+(10,20))$
190
Projective geometry
minimal embedding

$\OGr_+(10,20)\hookrightarrow\mathbb{P}^{ 511 }$

degree
3973186258569120
Hilbert series
1, 512, 92378, 8599552, 493684828, 19443587072, 564243782648, 12698585350144, 230316871499560, 3468301123758080, 44403738019988700, 492702777286809600, 4813956334721220750, 41969353060603584000, 330179705217925745625, 2366591592871733760000, 15582757631775063356250, 94937148555693200640000, 538557219826602143895000, 2860474167121736116224000, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
not recorded
self-dual
no

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

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\OGr_+(10,20))$:

Quantum cohomology
small
generically semisimple [Chaput–Manivel–Perrin]

Minuscule/cominuscule case, including quadric realisations.

big
generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]

Convergent near the large-radius limit.

Homological projective duality

No Lefschetz decomposition recorded.

No HP-dual model recorded.

$k$-Fanoness

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \OGr_+(10,20) })\cdot Z>0$ for every nonzero effective $i$-dimensional cycle $Z$ and $1\leq i\leq 2$. [Araujo et al., Theorems 1.3 and 1.4]

References
  1. Pierre-Emmanuel Chaput; Laurent Manivel; Nicolas Perrin. Quantum cohomology of minuscule homogeneous spaces III : semi-simplicity and consequences. Canad. J. Math. 62 (2010), 1246–1263. arXiv:0710.1224.
  2. 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.
  3. 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.
  4. 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.