\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 $X^1(5)$

Y Z 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 } &= 6 \\ \mathrm{b}_{ 12 } &= 7 \\ \mathrm{b}_{ 14 } &= 8 \\ \mathrm{b}_{ 16 } &= 9 \\ \mathrm{b}_{ 18 } &= 10 \\ \mathrm{b}_{ 20 } &= 10 \\ \mathrm{b}_{ 22 } &= 10 \\ \mathrm{b}_{ 24 } &= 9 \\ \mathrm{b}_{ 26 } &= 8 \\ \mathrm{b}_{ 28 } &= 7 \\ \mathrm{b}_{ 30 } &= 6 \\ \mathrm{b}_{ 32 } &= 4 \\ \mathrm{b}_{ 34 } &= 3 \\ \mathrm{b}_{ 36 } &= 2 \\ \mathrm{b}_{ 38 } &= 1 \\ \mathrm{b}_{ 40 } &= 1 \end{align*}
Basic information
dimension
20
index
7
Euler characteristic
112
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 6$, $\mathrm{b}_{ 12 } = 7$, $\mathrm{b}_{ 14 } = 8$, $\mathrm{b}_{ 16 } = 9$, $\mathrm{b}_{ 18 } = 10$, $\mathrm{b}_{ 20 } = 10$, $\mathrm{b}_{ 22 } = 10$, $\mathrm{b}_{ 24 } = 9$, $\mathrm{b}_{ 26 } = 8$, $\mathrm{b}_{ 28 } = 7$, $\mathrm{b}_{ 30 } = 6$, $\mathrm{b}_{ 32 } = 4$, $\mathrm{b}_{ 34 } = 3$, $\mathrm{b}_{ 36 } = 2$, $\mathrm{b}_{ 38 } = 1$, $\mathrm{b}_{ 40 } = 1$
$G$
$\mathrm{SO}_{ 11 }$
$\dim G$
50
$\mathrm{Aut}^0(X^1(5))$
$G\ltimes U$
$\dim\mathrm{Aut}^0(X^1(5))$
75
Blowups and projections
role dimension codimension index
$Z=\OGr(5,11)$ Y Z Y Z Y Z Y Z Y Z unique closed $\mathrm{Aut}(X^1(5))$-orbit 15 5 10
$Y=\OGr(4,11)$ Y Z Y Z Y Z Y Z Y Z other closed $\mathrm{SO}_{ 11 }$ -orbit 18 2 6
\begin{equation} \xymatrix{ E_{ \OGr(5,11) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \OGr(5,11) } X^1(5) \ar@{->>}[r]^(.6){\mathbb{P}^{ 2 }} \ar@{->>}[d] & \href{/B5/4 }{ \OGr(4,11) } \\ \href{/B5/5 }{ \OGr(5,11) } \ar[r] & X^1(5) } \end{equation} \begin{equation} \xymatrix{ E_{ \OGr(4,11) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \OGr(4,11) } X^1(5) \ar@{->>}[r]^(.6){\mathbb{P}^{ 5 }} \ar@{->>}[d] & \href{/B5/5 }{ \OGr(5,11) } \\ \href{/B5/4 }{ \OGr(4,11) } \ar[r] & X^1(5) } \end{equation}
The exceptional divisor is the partial flag variety Y Z Y Z Y Z Y Z Y Z
$ E_{ \OGr(4,11) } \cong E_{ \OGr(5,11) } \cong \mathrm{ B }_{ 5 } / \mathrm{P}_{ 4, 5 } $
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(X^1(5))$. 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