\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 $\operatorname{SGr}(2,5)=X^3(2, 2)$

Y Z Y Z
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 8 } &= 1 \\ \mathrm{b}_{ 10 } &= 1 \end{align*}
Basic information
dimension
5
index
4
Euler characteristic
8
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$
$G$
$\mathrm{PSp}_{ 4 }$
$\dim G$
8
$\mathrm{Aut}^0(X^3(2, 2))$
$G\ltimes U$
$\dim\mathrm{Aut}^0(X^3(2, 2))$
12

Geometric description
zero locus of $\bigwedge^2\mathcal{U}^\vee$ in $\operatorname{Gr}(2,5)$.
Blowups and projections
role dimension codimension index
$Z=\mathbb{P}^{3}$ Y Z Y Z unique closed $\mathrm{Aut}(X^3(2, 2))$-orbit 3 2 4
$Y=\mathrm{{Q}}^{{3}}$ Y Z Y Z other closed $\mathrm{PSp}_{ 4 }$ -orbit 3 2 3
\begin{equation} \xymatrix{ E_{ \mathbb{P}^{3} } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \mathbb{P}^{3} } X^3(2, 2) \ar@{->>}[r]^(.6){\mathbb{P}^{ 2 }} \ar@{->>}[d] & \href{/C2/2 }{ \mathrm{{Q}}^{{3}} } \\ \href{/C2/1 }{ \mathbb{P}^{3} } \ar[r] & X^3(2, 2) } \end{equation} \begin{equation} \xymatrix{ E_{ \mathrm{{Q}}^{{3}} } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \mathrm{{Q}}^{{3}} } X^3(2, 2) \ar@{->>}[r]^(.6){\mathbb{P}^{ 2 }} \ar@{->>}[d] & \href{/C2/1 }{ \mathbb{P}^{3} } \\ \href{/C2/2 }{ \mathrm{{Q}}^{{3}} } \ar[r] & X^3(2, 2) } \end{equation}
The exceptional divisor is the partial flag variety Y Z Y Z
$ E_{ \mathrm{{Q}}^{{3}} } \cong E_{ \mathbb{P}^{3} } \cong \mathrm{ C }_{ 2 } / \mathrm{P}_{ 2, 1 } $
Exceptional collections
  • Pech constructed a full exceptional sequence in 2013, see MR2998953.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(X^3(2, 2))$ are given by:

Homological projective duality