\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.

F₄-horospherical variety $X^4$

Y Z Y Z Y Z Y Z
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 4 \\ \mathrm{b}_{ 8 } &= 5 \\ \mathrm{b}_{ 10 } &= 7 \\ \mathrm{b}_{ 12 } &= 9 \\ \mathrm{b}_{ 14 } &= 11 \\ \mathrm{b}_{ 16 } &= 12 \\ \mathrm{b}_{ 18 } &= 14 \\ \mathrm{b}_{ 20 } &= 15 \\ \mathrm{b}_{ 22 } &= 15 \\ \mathrm{b}_{ 24 } &= 15 \\ \mathrm{b}_{ 26 } &= 15 \\ \mathrm{b}_{ 28 } &= 14 \\ \mathrm{b}_{ 30 } &= 12 \\ \mathrm{b}_{ 32 } &= 11 \\ \mathrm{b}_{ 34 } &= 9 \\ \mathrm{b}_{ 36 } &= 7 \\ \mathrm{b}_{ 38 } &= 5 \\ \mathrm{b}_{ 40 } &= 4 \\ \mathrm{b}_{ 42 } &= 2 \\ \mathrm{b}_{ 44 } &= 1 \\ \mathrm{b}_{ 46 } &= 1 \end{align*}
Basic information
dimension
23
index
6
Euler characteristic
192
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 10 } = 7$, $\mathrm{b}_{ 12 } = 9$, $\mathrm{b}_{ 14 } = 11$, $\mathrm{b}_{ 16 } = 12$, $\mathrm{b}_{ 18 } = 14$, $\mathrm{b}_{ 20 } = 15$, $\mathrm{b}_{ 22 } = 15$, $\mathrm{b}_{ 24 } = 15$, $\mathrm{b}_{ 26 } = 15$, $\mathrm{b}_{ 28 } = 14$, $\mathrm{b}_{ 30 } = 12$, $\mathrm{b}_{ 32 } = 11$, $\mathrm{b}_{ 34 } = 9$, $\mathrm{b}_{ 36 } = 7$, $\mathrm{b}_{ 38 } = 5$, $\mathrm{b}_{ 40 } = 4$, $\mathrm{b}_{ 42 } = 2$, $\mathrm{b}_{ 44 } = 1$, $\mathrm{b}_{ 46 } = 1$
$G$
$\mathrm{F}_4$
$\dim G$
48
$\mathrm{Aut}^0(X^4)$
$G\ltimes U$
$\dim\mathrm{Aut}^0(X^4)$
72
Blowups and projections
role dimension codimension index
$Z=\mathrm{F}_{4}/\mathrm{P}_{3}$ Y Z Y Z Y Z Y Z unique closed $\mathrm{Aut}(X^4)$-orbit 20 3 7
$Y=\mathrm{F}_{4}/\mathrm{P}_{2}$ Y Z Y Z Y Z Y Z other closed $\mathrm{F}_4$ -orbit 20 3 5
\begin{equation} \xymatrix{ E_{ \mathrm{F}_{4}/\mathrm{P}_{3} } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \mathrm{F}_{4}/\mathrm{P}_{3} } X^4 \ar@{->>}[r]^(.6){\mathbb{P}^{ 3 }} \ar@{->>}[d] & \href{/F4/2 }{ \mathrm{F}_{4}/\mathrm{P}_{2} } \\ \href{/F4/3 }{ \mathrm{F}_{4}/\mathrm{P}_{3} } \ar[r] & X^4 } \end{equation} \begin{equation} \xymatrix{ E_{ \mathrm{F}_{4}/\mathrm{P}_{2} } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \mathrm{F}_{4}/\mathrm{P}_{2} } X^4 \ar@{->>}[r]^(.6){\mathbb{P}^{ 3 }} \ar@{->>}[d] & \href{/F4/3 }{ \mathrm{F}_{4}/\mathrm{P}_{3} } \\ \href{/F4/2 }{ \mathrm{F}_{4}/\mathrm{P}_{2} } \ar[r] & X^4 } \end{equation}
The exceptional divisor is the partial flag variety Y Z Y Z Y Z Y Z
$ E_{ \mathrm{F}_{4}/\mathrm{P}_{2} } \cong E_{ \mathrm{F}_{4}/\mathrm{P}_{3} } \cong \mathrm{ F }_{ 4 } / \mathrm{P}_{ 2, 3 } $
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(X^4)$. Will you be the first to construct one? Let us know if you do!

Quantum cohomology

The small quantum cohomology is not known to be (non-)semisimple.

The big quantum cohomology is not known yet to be generically semisimple.

Homological projective duality