\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

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(X^1(3))$ are given by:

Homological projective duality