\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.

Generalised Grassmannian of type F4/P1

Y Z Y Z Y Z Y Z
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 } &= 2 \\ \mathrm{b}_{ 18 } &= 2 \\ \mathrm{b}_{ 20 } &= 2 \\ \mathrm{b}_{ 22 } &= 2 \\ \mathrm{b}_{ 24 } &= 1 \\ \mathrm{b}_{ 26 } &= 1 \\ \mathrm{b}_{ 28 } &= 1 \\ \mathrm{b}_{ 30 } &= 1 \end{align*}
Basic information
dimension
15
index
8
Euler characteristic
24
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 2$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 20 } = 2$, $\mathrm{b}_{ 22 } = 2$, $\mathrm{b}_{ 24 } = 1$, $\mathrm{b}_{ 26 } = 1$, $\mathrm{b}_{ 28 } = 1$, $\mathrm{b}_{ 30 } = 1$
$\mathrm{Aut}^0(\mathrm{F}_{4}/\mathrm{P}_{1})$
$\mathrm{F}_4$
$\pi_0\mathrm{Aut}(\mathrm{F}_{4}/\mathrm{P}_{1})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{F}_{4}/\mathrm{P}_{1})$
52
Projective geometry
minimal embedding

$\mathrm{F}_{4}/\mathrm{P}_{1}\hookrightarrow\mathbb{P}^{ 51 }$

degree
4992
Hilbert series
1, 52, 1053, 12376, 100776, 627912, 3187041, 13748020, 51949755, 175847880, 542393670, 1544927904, 4107092288, 10278624864, 24388573014, 55188666312, 119696471453, 249869263644, 503865726155, 984563860280, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
24
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 $\mathbf D^{\mathrm b}(\mathrm{F}_{4}/\mathrm{P}_{1})$:

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.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{F}_{4}/\mathrm{P}_{1})$:

Homological projective duality

Full minimal rectangular Lefschetz collection. [Smirnov, Theorem 1.1]

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

Block sizes $(3, 3, 3, 3, 3, 3, 3, 3)$; 24 objects.

The structure sheaf, a homogeneous bundle, and the extension of the tangent bundle by the structure sheaf.

No HP-dual model recorded.

$k$-Fanoness

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

References
  1. 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.
  2. Maxim Smirnov. On the derived category of the adjoint Grassmannian of type F. Accepted for Algebraic Geometry and Physics (volume in honour of Yuri I. Manin); preprint revised 2026. arXiv:2107.07814.
  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.