Grassmannian.info

dimension

9

index

7

Euler characteristic

14

Betti numbers

$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 14 } = 1$, $\mathrm{b}_{ 16 } = 1$, $\mathrm{b}_{ 18 } = 1$

$G$

$\mathrm{SO}_{ 7 }$

$\dim G$

18

$\mathrm{Aut}^0(X^2)$

$G\ltimes U$

$\dim\mathrm{Aut}^0(X^2)$

27

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Geometric description

hyperplane section of $\operatorname{OGr}_+(5,10)$.

	role	dimension	codimension	index
$Z=\OGr(3,7)$	unique closed $\mathrm{Aut}(X^2)$-orbit	6	3	6
$Y=\mathrm{Q}^{5}$	other closed $\mathrm{SO}_{ 7 }$ -orbit	5	4	5

\begin{equation} \xymatrix{ E_{ \OGr(3,7) } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \OGr(3,7) } X^2 \ar@{->>}[r]^(.6){\mathbb{P}^{ 4 }} \ar@{->>}[d] & \href{/B3/1 }{ \mathrm{Q}^{5} } \\ \href{/B3/3 }{ \OGr(3,7) } \ar[r] & X^2 } \end{equation} \begin{equation} \xymatrix{ E_{ \mathrm{Q}^{5} } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \mathrm{Q}^{5} } X^2 \ar@{->>}[r]^(.6){\mathbb{P}^{ 3 }} \ar@{->>}[d] & \href{/B3/3 }{ \OGr(3,7) } \\ \href{/B3/1 }{ \mathrm{Q}^{5} } \ar[r] & X^2 } \end{equation}

• Gonzales–Pech–Perrin–Samokhin constructed a full exceptional sequence in 2018, see Gonzales–Pech–Perrin–Samokhin (2022) .

small

generically semisimple

big

generically semisimple

Eigenvalues of quantum multiplication by $\mathrm{c}_1(X^2)$: