\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.

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

Y Z 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 } &= 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 \end{align*}
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
$Z=\OGr(7,15)$ Y Z Y Z Y Z Y Z Y Z Y Z Y Z unique closed $\mathrm{Aut}(X^1(7))$-orbit 28 7 14
$Y=\OGr(6,15)$ Y Z Y Z Y Z Y Z Y Z Y Z Y Z other closed $\mathrm{SO}_{ 15 }$ -orbit 33 2 8
\begin{equation} \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) } \end{equation} \begin{equation} \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) } \end{equation}
The exceptional divisor is the partial flag variety Y Z Y Z Y Z Y Z Y Z Y Z Y Z
$ 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