# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Horo-orthogonal Grassmannian $X^1(7)$

Basic information
dimension
35
index
9
Euler characteristic
576
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 6$, $\mathrm{b}_{ 12 } = 8$, $\mathrm{b}_{ 14 } = 11$, $\mathrm{b}_{ 16 } = 13$, $\mathrm{b}_{ 18 } = 16$, $\mathrm{b}_{ 20 } = 19$, $\mathrm{b}_{ 22 } = 22$, $\mathrm{b}_{ 24 } = 25$, $\mathrm{b}_{ 26 } = 28$, $\mathrm{b}_{ 28 } = 30$, $\mathrm{b}_{ 30 } = 32$, $\mathrm{b}_{ 32 } = 33$, $\mathrm{b}_{ 34 } = 34$, $\mathrm{b}_{ 36 } = 34$, $\mathrm{b}_{ 38 } = 33$, $\mathrm{b}_{ 40 } = 32$, $\mathrm{b}_{ 42 } = 30$, $\mathrm{b}_{ 44 } = 28$, $\mathrm{b}_{ 46 } = 25$, $\mathrm{b}_{ 48 } = 22$, $\mathrm{b}_{ 50 } = 19$, $\mathrm{b}_{ 52 } = 16$, $\mathrm{b}_{ 54 } = 13$, $\mathrm{b}_{ 56 } = 11$, $\mathrm{b}_{ 58 } = 8$, $\mathrm{b}_{ 60 } = 6$, $\mathrm{b}_{ 62 } = 4$, $\mathrm{b}_{ 64 } = 3$, $\mathrm{b}_{ 66 } = 2$, $\mathrm{b}_{ 68 } = 1$, $\mathrm{b}_{ 70 } = 1$
$G$
$\mathrm{SO}_{ 15 }$
$\dim G$
98
$\mathrm{Aut}^0(X^1(7))$
$G\ltimes U$
$\dim\mathrm{Aut}^0(X^1(7))$
147
Blowups and projections
role dimension codimension index
$Z=\OGr(7,15)$ unique closed $\mathrm{Aut}(X^1(7))$-orbit 28 7 14
$Y=\OGr(6,15)$ other closed $\mathrm{SO}_{ 15 }$ -orbit 33 2 8
$$\xymatrix{ E_{ \OGr(7,15) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \OGr(7,15) } X^1(7) \ar@{->>}[r]^(.6){\mathbb{P}^{ 2 }} \ar@{->>}[d] & \href{/B7/6 }{ \OGr(6,15) } \\ \href{/B7/7 }{ \OGr(7,15) } \ar[r] & X^1(7) }$$ $$\xymatrix{ E_{ \OGr(6,15) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \OGr(6,15) } X^1(7) \ar@{->>}[r]^(.6){\mathbb{P}^{ 7 }} \ar@{->>}[d] & \href{/B7/7 }{ \OGr(7,15) } \\ \href{/B7/6 }{ \OGr(6,15) } \ar[r] & X^1(7) }$$
The exceptional divisor is the partial flag variety
$E_{ \OGr(6,15) } \cong E_{ \OGr(7,15) } \cong \mathrm{ B }_{ 7 } / \mathrm{P}_{ 6, 7 }$
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(X^1(7))$. Will you be the first to construct one? Let us know if you do!

Quantum cohomology

The small quantum cohomology is not known to be (non-)semisimple.

The big quantum cohomology is not known yet to be generically semisimple.

Homological projective duality