Grassmannian.info

A periodic table of (generalised) Grassmannians.

Odd symplectic Grassmannian $\operatorname{SGr}(5,15)=X^3(7, 5)$

Basic information
dimension
40
index
11
Euler characteristic
1232
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 10 } = 7$, $\mathrm{b}_{ 12 } = 10$, $\mathrm{b}_{ 14 } = 13$, $\mathrm{b}_{ 16 } = 17$, $\mathrm{b}_{ 18 } = 22$, $\mathrm{b}_{ 20 } = 27$, $\mathrm{b}_{ 22 } = 32$, $\mathrm{b}_{ 24 } = 38$, $\mathrm{b}_{ 26 } = 44$, $\mathrm{b}_{ 28 } = 49$, $\mathrm{b}_{ 30 } = 55$, $\mathrm{b}_{ 32 } = 59$, $\mathrm{b}_{ 34 } = 63$, $\mathrm{b}_{ 36 } = 66$, $\mathrm{b}_{ 38 } = 68$, $\mathrm{b}_{ 40 } = 68$, $\mathrm{b}_{ 42 } = 68$, $\mathrm{b}_{ 44 } = 66$, $\mathrm{b}_{ 46 } = 63$, $\mathrm{b}_{ 48 } = 59$, $\mathrm{b}_{ 50 } = 55$, $\mathrm{b}_{ 52 } = 49$, $\mathrm{b}_{ 54 } = 44$, $\mathrm{b}_{ 56 } = 38$, $\mathrm{b}_{ 58 } = 32$, $\mathrm{b}_{ 60 } = 27$, $\mathrm{b}_{ 62 } = 22$, $\mathrm{b}_{ 64 } = 17$, $\mathrm{b}_{ 66 } = 13$, $\mathrm{b}_{ 68 } = 10$, $\mathrm{b}_{ 70 } = 7$, $\mathrm{b}_{ 72 } = 5$, $\mathrm{b}_{ 74 } = 3$, $\mathrm{b}_{ 76 } = 2$, $\mathrm{b}_{ 78 } = 1$, $\mathrm{b}_{ 80 } = 1$
$G$
$\mathrm{PSp}_{ 14 }$
$\dim G$
98
$\mathrm{Aut}^0(X^3(7, 5))$
$G\ltimes U$
$\dim\mathrm{Aut}^0(X^3(7, 5))$
147

Geometric description
zero locus of $\bigwedge^2\mathcal{U}^\vee$ in $\operatorname{Gr}(5,15)$.
Blowups and projections
role dimension codimension index
$Z=\SGr(4,14)$ unique closed $\mathrm{Aut}(X^3(7, 5))$-orbit 34 6 11
$Y=\SGr(5,14)$ other closed $\mathrm{PSp}_{ 14 }$ -orbit 35 5 10
$$\xymatrix{ E_{ \SGr(4,14) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \SGr(4,14) } X^3(7, 5) \ar@{->>}[r]^(.6){\mathbb{P}^{ 5 }} \ar@{->>}[d] & \href{/C7/5 }{ \SGr(5,14) } \\ \href{/C7/4 }{ \SGr(4,14) } \ar[r] & X^3(7, 5) }$$ $$\xymatrix{ E_{ \SGr(5,14) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \SGr(5,14) } X^3(7, 5) \ar@{->>}[r]^(.6){\mathbb{P}^{ 6 }} \ar@{->>}[d] & \href{/C7/4 }{ \SGr(4,14) } \\ \href{/C7/5 }{ \SGr(5,14) } \ar[r] & X^3(7, 5) }$$
The exceptional divisor is the partial flag variety
$E_{ \SGr(5,14) } \cong E_{ \SGr(4,14) } \cong \mathrm{ C }_{ 7 } / \mathrm{P}_{ 5, 4 }$
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(X^3(7, 5))$. 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 $\mathrm{c}_1(X^3(7, 5))$ are given by:

Homological projective duality