\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.

Horo-orthogonal Grassmannian $X^1(4)$

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 } &= 3 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 10 } &= 5 \\ \mathrm{b}_{ 12 } &= 5 \\ \mathrm{b}_{ 14 } &= 6 \\ \mathrm{b}_{ 16 } &= 5 \\ \mathrm{b}_{ 18 } &= 5 \\ \mathrm{b}_{ 20 } &= 4 \\ \mathrm{b}_{ 22 } &= 3 \\ \mathrm{b}_{ 24 } &= 2 \\ \mathrm{b}_{ 26 } &= 1 \\ \mathrm{b}_{ 28 } &= 1 \end{align*}
Basic information
dimension
14
index
6
Euler characteristic
48
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 12 } = 5$, $\mathrm{b}_{ 14 } = 6$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 18 } = 5$, $\mathrm{b}_{ 20 } = 4$, $\mathrm{b}_{ 22 } = 3$, $\mathrm{b}_{ 24 } = 2$, $\mathrm{b}_{ 26 } = 1$, $\mathrm{b}_{ 28 } = 1$
$G$
$\mathrm{SO}_{ 9 }$
$\dim G$
32
$\mathrm{Aut}^0(X^1(4))$
$G\ltimes U$
$\dim\mathrm{Aut}^0(X^1(4))$
48
Blowups and projections
role dimension codimension index
$Z=\OGr(4,9)$ Y Z Y Z Y Z Y Z unique closed $\mathrm{Aut}(X^1(4))$-orbit 10 4 8
$Y=\OGr(3,9)$ Y Z Y Z Y Z Y Z other closed $\mathrm{SO}_{ 9 }$ -orbit 12 2 5
\begin{equation} \xymatrix{ E_{ \OGr(4,9) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \OGr(4,9) } X^1(4) \ar@{->>}[r]^(.6){\mathbb{P}^{ 2 }} \ar@{->>}[d] & \href{/B4/3 }{ \OGr(3,9) } \\ \href{/B4/4 }{ \OGr(4,9) } \ar[r] & X^1(4) } \end{equation} \begin{equation} \xymatrix{ E_{ \OGr(3,9) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \OGr(3,9) } X^1(4) \ar@{->>}[r]^(.6){\mathbb{P}^{ 4 }} \ar@{->>}[d] & \href{/B4/4 }{ \OGr(4,9) } \\ \href{/B4/3 }{ \OGr(3,9) } \ar[r] & X^1(4) } \end{equation}
The exceptional divisor is the partial flag variety Y Z Y Z Y Z Y Z
$ E_{ \OGr(3,9) } \cong E_{ \OGr(4,9) } \cong \mathrm{ B }_{ 4 } / \mathrm{P}_{ 3, 4 } $
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(X^1(4))$. 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