\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.

Cayley plane $\mathbb{OP}^2$

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
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 3$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 20 } = 2$, $\mathrm{b}_{ 22 } = 2$, $\mathrm{b}_{ 24 } = 2$, $\mathrm{b}_{ 26 } = 1$, $\mathrm{b}_{ 28 } = 1$, $\mathrm{b}_{ 30 } = 1$, $\mathrm{b}_{ 32 } = 1$
$\mathrm{Aut}^0(\mathbb{OP}^2)$
adjoint group of type $\mathrm{E}_{ 6 }$
$\pi_0\mathrm{Aut}(\mathbb{OP}^2)$
$1$
$\dim\mathrm{Aut}^0(\mathbb{OP}^2)$
78
Projective geometry
minimal embedding

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

degree
78
Hilbert series
1, 27, 351, 3003, 19305, 100386, 442442, 1706562, 5895396, 18559580, 53965548, 146477916, 374332452, 907036326, 2096092350, 4642456390, 9895762305, 20373628275, 40639459575, 78751105875, ...
Exceptional collections
  • Faenzi–Manivel constructed a full exceptional sequence in 2013, see MR3293722.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathbb{OP}^2)$ are given by:

Homological projective duality