\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 $X^1(3)$

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 } &= 3 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 12 } &= 3 \\ \mathrm{b}_{ 14 } &= 2 \\ \mathrm{b}_{ 16 } &= 1 \\ \mathrm{b}_{ 18 } &= 1 \end{align*}
Basic information
dimension
9
index
5
Euler characteristic
20
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 3$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 1$, $\mathrm{b}_{ 18 } = 1$
$G$
$\mathrm{SO}_{ 7 }$
$\dim G$
18
$\mathrm{Aut}^0(X^1(3))$
$G\ltimes U$
$\dim\mathrm{Aut}^0(X^1(3))$
27
Blowups and projections
role dimension codimension index
$Z=\OGr(3,7)$ Y Z Y Z Y Z unique closed $\mathrm{Aut}(X^1(3))$-orbit 6 3 6
$Y=\OGr(2,7)$ Y Z Y Z Y Z other closed $\mathrm{SO}_{ 7 }$ -orbit 7 2 4
\begin{equation} \xymatrix{ E_{ \OGr(3,7) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \OGr(3,7) } X^1(3) \ar@{->>}[r]^(.6){\mathbb{P}^{ 2 }} \ar@{->>}[d] & \href{/B3/2 }{ \OGr(2,7) } \\ \href{/B3/3 }{ \OGr(3,7) } \ar[r] & X^1(3) } \end{equation} \begin{equation} \xymatrix{ E_{ \OGr(2,7) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \OGr(2,7) } X^1(3) \ar@{->>}[r]^(.6){\mathbb{P}^{ 3 }} \ar@{->>}[d] & \href{/B3/3 }{ \OGr(3,7) } \\ \href{/B3/2 }{ \OGr(2,7) } \ar[r] & X^1(3) } \end{equation}
The exceptional divisor is the partial flag variety Y Z Y Z Y Z
$ E_{ \OGr(2,7) } \cong E_{ \OGr(3,7) } \cong \mathrm{ B }_{ 3 } / \mathrm{P}_{ 2, 3 } $
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(X^1(3))$. Will you be the first to construct one? Let us know if you do!

Quantum cohomology
small
generically semisimple
big
generically semisimple

Eigenvalues of quantum multiplication by $\mathrm{c}_1(X^1(3))$:

Homological projective duality