\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.

Odd symplectic Grassmannian $\operatorname{SGr}(3,9)=X^3(4, 3)$

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 } &= 6 \\ \mathrm{b}_{ 14 } &= 6 \\ \mathrm{b}_{ 16 } &= 6 \\ \mathrm{b}_{ 18 } &= 6 \\ \mathrm{b}_{ 20 } &= 5 \\ \mathrm{b}_{ 22 } &= 4 \\ \mathrm{b}_{ 24 } &= 3 \\ \mathrm{b}_{ 26 } &= 2 \\ \mathrm{b}_{ 28 } &= 1 \\ \mathrm{b}_{ 30 } &= 1 \end{align*}
Basic information
dimension
15
index
7
Euler characteristic
56
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 12 } = 6$, $\mathrm{b}_{ 14 } = 6$, $\mathrm{b}_{ 16 } = 6$, $\mathrm{b}_{ 18 } = 6$, $\mathrm{b}_{ 20 } = 5$, $\mathrm{b}_{ 22 } = 4$, $\mathrm{b}_{ 24 } = 3$, $\mathrm{b}_{ 26 } = 2$, $\mathrm{b}_{ 28 } = 1$, $\mathrm{b}_{ 30 } = 1$
$G$
$\mathrm{PSp}_{ 8 }$
$\dim G$
32
$\mathrm{Aut}^0(X^3(4, 3))$
$G\ltimes U$
$\dim\mathrm{Aut}^0(X^3(4, 3))$
48

Geometric description
zero locus of $\bigwedge^2\mathcal{U}^\vee$ in $\operatorname{Gr}(3,9)$.
Blowups and projections
role dimension codimension index
$Z=\SGr(2,8)$ Y Z Y Z Y Z Y Z unique closed $\mathrm{Aut}(X^3(4, 3))$-orbit 11 4 7
$Y=\SGr(3,8)$ Y Z Y Z Y Z Y Z other closed $\mathrm{PSp}_{ 8 }$ -orbit 12 3 6
\begin{equation} \xymatrix{ E_{ \SGr(2,8) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \SGr(2,8) } X^3(4, 3) \ar@{->>}[r]^(.6){\mathbb{P}^{ 3 }} \ar@{->>}[d] & \href{/C4/3 }{ \SGr(3,8) } \\ \href{/C4/2 }{ \SGr(2,8) } \ar[r] & X^3(4, 3) } \end{equation} \begin{equation} \xymatrix{ E_{ \SGr(3,8) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \SGr(3,8) } X^3(4, 3) \ar@{->>}[r]^(.6){\mathbb{P}^{ 4 }} \ar@{->>}[d] & \href{/C4/2 }{ \SGr(2,8) } \\ \href{/C4/3 }{ \SGr(3,8) } \ar[r] & X^3(4, 3) } \end{equation}
The exceptional divisor is the partial flag variety Y Z Y Z Y Z Y Z
$ E_{ \SGr(3,8) } \cong E_{ \SGr(2,8) } \cong \mathrm{ C }_{ 4 } / \mathrm{P}_{ 3, 2 } $
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(X^3(4, 3))$. 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 $\mathrm{c}_1(X^3(4, 3))$ are given by:

Homological projective duality