\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 $\operatorname{SGr}(3,13)=X^3(6, 3)$

Y Z 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 } &= 5 \\ \mathrm{b}_{ 12 } &= 7 \\ \mathrm{b}_{ 14 } &= 8 \\ \mathrm{b}_{ 16 } &= 10 \\ \mathrm{b}_{ 18 } &= 12 \\ \mathrm{b}_{ 20 } &= 13 \\ \mathrm{b}_{ 22 } &= 14 \\ \mathrm{b}_{ 24 } &= 15 \\ \mathrm{b}_{ 26 } &= 15 \\ \mathrm{b}_{ 28 } &= 15 \\ \mathrm{b}_{ 30 } &= 15 \\ \mathrm{b}_{ 32 } &= 14 \\ \mathrm{b}_{ 34 } &= 13 \\ \mathrm{b}_{ 36 } &= 12 \\ \mathrm{b}_{ 38 } &= 10 \\ \mathrm{b}_{ 40 } &= 8 \\ \mathrm{b}_{ 42 } &= 7 \\ \mathrm{b}_{ 44 } &= 5 \\ \mathrm{b}_{ 46 } &= 4 \\ \mathrm{b}_{ 48 } &= 3 \\ \mathrm{b}_{ 50 } &= 2 \\ \mathrm{b}_{ 52 } &= 1 \\ \mathrm{b}_{ 54 } &= 1 \end{align*}
Basic information
dimension
27
index
11
Euler characteristic
220
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 12 } = 7$, $\mathrm{b}_{ 14 } = 8$, $\mathrm{b}_{ 16 } = 10$, $\mathrm{b}_{ 18 } = 12$, $\mathrm{b}_{ 20 } = 13$, $\mathrm{b}_{ 22 } = 14$, $\mathrm{b}_{ 24 } = 15$, $\mathrm{b}_{ 26 } = 15$, $\mathrm{b}_{ 28 } = 15$, $\mathrm{b}_{ 30 } = 15$, $\mathrm{b}_{ 32 } = 14$, $\mathrm{b}_{ 34 } = 13$, $\mathrm{b}_{ 36 } = 12$, $\mathrm{b}_{ 38 } = 10$, $\mathrm{b}_{ 40 } = 8$, $\mathrm{b}_{ 42 } = 7$, $\mathrm{b}_{ 44 } = 5$, $\mathrm{b}_{ 46 } = 4$, $\mathrm{b}_{ 48 } = 3$, $\mathrm{b}_{ 50 } = 2$, $\mathrm{b}_{ 52 } = 1$, $\mathrm{b}_{ 54 } = 1$
$G$
$\mathrm{PSp}_{ 12 }$
$\dim G$
72
$\mathrm{Aut}^0(X^3(6, 3))$
$G\ltimes U$
$\dim\mathrm{Aut}^0(X^3(6, 3))$
108
Geometric description
zero locus of $\bigwedge^2\mathcal{U}^\vee$ in $\operatorname{Gr}(3,13)$.
Blowups and projections
role dimension codimension index
$Z=\SGr(2,12)$ Y Z Y Z Y Z Y Z Y Z Y Z unique closed $\mathrm{Aut}(X^3(6, 3))$-orbit 19 8 11
$Y=\SGr(3,12)$ Y Z Y Z Y Z Y Z Y Z Y Z other closed $\mathrm{PSp}_{ 12 }$ -orbit 24 3 10
\begin{equation} \xymatrix{ E_{ \SGr(2,12) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \SGr(2,12) } X^3(6, 3) \ar@{->>}[r]^(.6){\mathbb{P}^{ 3 }} \ar@{->>}[d] & \href{/C6/3 }{ \SGr(3,12) } \\ \href{/C6/2 }{ \SGr(2,12) } \ar[r] & X^3(6, 3) } \end{equation} \begin{equation} \xymatrix{ E_{ \SGr(3,12) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \SGr(3,12) } X^3(6, 3) \ar@{->>}[r]^(.6){\mathbb{P}^{ 8 }} \ar@{->>}[d] & \href{/C6/2 }{ \SGr(2,12) } \\ \href{/C6/3 }{ \SGr(3,12) } \ar[r] & X^3(6, 3) } \end{equation}
The exceptional divisor is the partial flag variety Y Z Y Z Y Z Y Z Y Z Y Z
$ E_{ \SGr(3,12) } \cong E_{ \SGr(2,12) } \cong \mathrm{ C }_{ 6 } / \mathrm{P}_{ 3, 2 } $
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(X^3(6, 3))$. 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 $\mathrm{c}_1(X^3(6, 3))$:

Homological projective duality