OGr+(12,24)—grassmannian.info

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 } &= 10 \\ \mathrm{b}_{ 22 } &= 12 \\ \mathrm{b}_{ 24 } &= 14 \\ \mathrm{b}_{ 26 } &= 16 \\ \mathrm{b}_{ 28 } &= 19 \\ \mathrm{b}_{ 30 } &= 22 \\ \mathrm{b}_{ 32 } &= 25 \\ \mathrm{b}_{ 34 } &= 28 \\ \mathrm{b}_{ 36 } &= 32 \\ \mathrm{b}_{ 38 } &= 35 \\ \mathrm{b}_{ 40 } &= 39 \\ \mathrm{b}_{ 42 } &= 43 \\ \mathrm{b}_{ 44 } &= 46 \\ \mathrm{b}_{ 46 } &= 49 \\ \mathrm{b}_{ 48 } &= 53 \\ \mathrm{b}_{ 50 } &= 56 \\ \mathrm{b}_{ 52 } &= 59 \\ \mathrm{b}_{ 54 } &= 62 \\ \mathrm{b}_{ 56 } &= 64 \\ \mathrm{b}_{ 58 } &= 66 \\ \mathrm{b}_{ 60 } &= 68 \\ \mathrm{b}_{ 62 } &= 69 \\ \mathrm{b}_{ 64 } &= 69 \\ \mathrm{b}_{ 66 } &= 70 \\ \mathrm{b}_{ 68 } &= 69 \\ \mathrm{b}_{ 70 } &= 69 \\ \mathrm{b}_{ 72 } &= 68 \\ \mathrm{b}_{ 74 } &= 66 \\ \mathrm{b}_{ 76 } &= 64 \\ \mathrm{b}_{ 78 } &= 62 \\ \mathrm{b}_{ 80 } &= 59 \\ \mathrm{b}_{ 82 } &= 56 \\ \mathrm{b}_{ 84 } &= 53 \\ \mathrm{b}_{ 86 } &= 49 \\ \mathrm{b}_{ 88 } &= 46 \\ \mathrm{b}_{ 90 } &= 43 \\ \mathrm{b}_{ 92 } &= 39 \\ \mathrm{b}_{ 94 } &= 35 \\ \mathrm{b}_{ 96 } &= 32 \\ \mathrm{b}_{ 98 } &= 28 \\ \mathrm{b}_{ 100 } &= 25 \\ \mathrm{b}_{ 102 } &= 22 \\ \mathrm{b}_{ 104 } &= 19 \\ \mathrm{b}_{ 106 } &= 16 \\ \mathrm{b}_{ 108 } &= 14 \\ \mathrm{b}_{ 110 } &= 12 \\ \mathrm{b}_{ 112 } &= 10 \\ \mathrm{b}_{ 114 } &= 8 \\ \mathrm{b}_{ 116 } &= 6 \\ \mathrm{b}_{ 118 } &= 5 \\ \mathrm{b}_{ 120 } &= 4 \\ \mathrm{b}_{ 122 } &= 3 \\ \mathrm{b}_{ 124 } &= 2 \\ \mathrm{b}_{ 126 } &= 2 \\ \mathrm{b}_{ 128 } &= 1 \\ \mathrm{b}_{ 130 } &= 1 \\ \mathrm{b}_{ 132 } &= 1 \end{align*}

Basic information

dimension: 66
index: 22
Euler characteristic: 2048
Betti numbers: $\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 5$, $\mathrm{b}_{ 16 } = 6$, $\mathrm{b}_{ 18 } = 8$, $\mathrm{b}_{ 20 } = 10$, $\mathrm{b}_{ 22 } = 12$, $\mathrm{b}_{ 24 } = 14$, $\mathrm{b}_{ 26 } = 16$, $\mathrm{b}_{ 28 } = 19$, $\mathrm{b}_{ 30 } = 22$, $\mathrm{b}_{ 32 } = 25$, $\mathrm{b}_{ 34 } = 28$, $\mathrm{b}_{ 36 } = 32$, $\mathrm{b}_{ 38 } = 35$, $\mathrm{b}_{ 40 } = 39$, $\mathrm{b}_{ 42 } = 43$, $\mathrm{b}_{ 44 } = 46$, $\mathrm{b}_{ 46 } = 49$, $\mathrm{b}_{ 48 } = 53$, $\mathrm{b}_{ 50 } = 56$, $\mathrm{b}_{ 52 } = 59$, $\mathrm{b}_{ 54 } = 62$, $\mathrm{b}_{ 56 } = 64$, $\mathrm{b}_{ 58 } = 66$, $\mathrm{b}_{ 60 } = 68$, $\mathrm{b}_{ 62 } = 69$, $\mathrm{b}_{ 64 } = 69$, $\mathrm{b}_{ 66 } = 70$, $\mathrm{b}_{ 68 } = 69$, $\mathrm{b}_{ 70 } = 69$, $\mathrm{b}_{ 72 } = 68$, $\mathrm{b}_{ 74 } = 66$, $\mathrm{b}_{ 76 } = 64$, $\mathrm{b}_{ 78 } = 62$, $\mathrm{b}_{ 80 } = 59$, $\mathrm{b}_{ 82 } = 56$, $\mathrm{b}_{ 84 } = 53$, $\mathrm{b}_{ 86 } = 49$, $\mathrm{b}_{ 88 } = 46$, $\mathrm{b}_{ 90 } = 43$, $\mathrm{b}_{ 92 } = 39$, $\mathrm{b}_{ 94 } = 35$, $\mathrm{b}_{ 96 } = 32$, $\mathrm{b}_{ 98 } = 28$, $\mathrm{b}_{ 100 } = 25$, $\mathrm{b}_{ 102 } = 22$, $\mathrm{b}_{ 104 } = 19$, $\mathrm{b}_{ 106 } = 16$, $\mathrm{b}_{ 108 } = 14$, $\mathrm{b}_{ 110 } = 12$, $\mathrm{b}_{ 112 } = 10$, $\mathrm{b}_{ 114 } = 8$, $\mathrm{b}_{ 116 } = 6$, $\mathrm{b}_{ 118 } = 5$, $\mathrm{b}_{ 120 } = 4$, $\mathrm{b}_{ 122 } = 3$, $\mathrm{b}_{ 124 } = 2$, $\mathrm{b}_{ 126 } = 2$, $\mathrm{b}_{ 128 } = 1$, $\mathrm{b}_{ 130 } = 1$, $\mathrm{b}_{ 132 } = 1$
$\mathrm{Aut}^0(\OGr_+(12,24))$: $\mathrm{PSO}_{ 24 }$
$\pi_0\mathrm{Aut}(\OGr_+(12,24))$: $1$
$\dim\mathrm{Aut}^0(\OGr_+(12,24))$: 276

Projective geometry

minimal embedding: $\OGr_+(12,24)\hookrightarrow\mathbb{P}^{ 2047 }$
degree: 3830793890438041335187545600
Hilbert series: 1, 2048, 1352078, 426008576, 77266057400, 9042336317440, 738012266260200, 44454385920614400, 2063124143451951750, 76307914655642880000, 2311257936525480219375, 58618653300361405440000, 1268196736493693617762500, 23773610218206723379200000, 391341716055319415000625000, 5722150214145624222412800000, 75064857535224682029605925000, 891204801345739993023959040000, 9649786883842236128891910660000, 95942594459299291229671096320000, ...

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 $\mathbf D^{\mathrm b}(\OGr_+(12,24))$:

Samokhin–van der Kallen (2024)

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: $\mathrm{ch}_i(\mathrm{T}_{ \OGr_+(12,24) })\cdot Z>0$ for every nonzero effective $i$-dimensional cycle $Z$ and $1\leq i\leq 2$. [Araujo et al., Theorems 1.3 and 1.4]

References

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.
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, spinor variety $\OGr_+(12,24)$