\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.

Dual Cayley plane $\mathbb{OP}^{2,\vee}$

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 } &= 1 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 10 } &= 2 \\ \mathrm{b}_{ 12 } &= 2 \\ \mathrm{b}_{ 14 } &= 2 \\ \mathrm{b}_{ 16 } &= 3 \\ \mathrm{b}_{ 18 } &= 2 \\ \mathrm{b}_{ 20 } &= 2 \\ \mathrm{b}_{ 22 } &= 2 \\ \mathrm{b}_{ 24 } &= 2 \\ \mathrm{b}_{ 26 } &= 1 \\ \mathrm{b}_{ 28 } &= 1 \\ \mathrm{b}_{ 30 } &= 1 \\ \mathrm{b}_{ 32 } &= 1 \end{align*}
Basic information
dimension
16
index
12
Euler characteristic
27
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 3$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 20 } = 2$, $\mathrm{b}_{ 22 } = 2$, $\mathrm{b}_{ 24 } = 2$, $\mathrm{b}_{ 26 } = 1$, $\mathrm{b}_{ 28 } = 1$, $\mathrm{b}_{ 30 } = 1$, $\mathrm{b}_{ 32 } = 1$
$\mathrm{Aut}^0(\mathbb{OP}^{2,\vee})$
adjoint group of type $\mathrm{E}_{ 6 }$
$\pi_0\mathrm{Aut}(\mathbb{OP}^{2,\vee})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{OP}^{2,\vee})$
78
Projective geometry
minimal embedding

$\mathbb{OP}^{2,\vee}\hookrightarrow\mathbb{P}^{ 26 }$

degree
78
Hilbert series
1, 27, 351, 3003, 19305, 100386, 442442, 1706562, 5895396, 18559580, 53965548, 146477916, 374332452, 907036326, 2096092350, 4642456390, 9895762305, 20373628275, 40639459575, 78751105875, ...
Projective duality

The Cartan cubic hypersurface.

codimension
1
degree
3
self-dual
no

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] [Landsberg–Manivel, Section 4.1] All Grassmannians.

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\mathbb{OP}^{2,\vee})$:

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.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathbb{OP}^{2,\vee})$:

Homological projective duality

Full minimal nonrectangular Lefschetz collection. [Faenzi–Manivel, Theorem 1]

Lefschetz block sizes: 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2

Block sizes $(3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2)$; 27 objects.

Three homogeneous bundles in the first three blocks; two in the remaining blocks.

No HP-dual model recorded.

$k$-Fanoness

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \mathbb{OP}^{2,\vee} })\cdot Z>0$ 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. Faenzi, Daniele; Manivel, Laurent. On the derived category of the Cayley plane II. Proc. Amer. Math. Soc. 143 (2015), 1057–1074. MR3293722.
  3. 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.
  4. 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.
  5. 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.