X5—grassmannian.info

Betti numbers

\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 10 } &= 2 \\ \mathrm{b}_{ 12 } &= 1 \\ \mathrm{b}_{ 14 } &= 1 \end{align*}

Basic information

dimension: 7
index: 4
Euler characteristic: 12
Betti numbers: $\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 14 } = 1$
$G$: $\mathrm{G}_2$
$\dim G$: 12
$\mathrm{Aut}^0(X^5)$: $G\ltimes U$
$\dim\mathrm{Aut}^0(X^5)$: 18

Blowups and projections

	role	dimension	codimension	index
$Z=\mathrm{G}_{2}/\mathrm{P}_{2}$	unique closed $\mathrm{Aut}(X^5)$-orbit	5	2	3
$Y=\mathrm{Q}^{5}$	other closed $\mathrm{G}_2$ -orbit	5	2	5

\begin{equation} \xymatrix{ E_{ \mathrm{G}_{2}/\mathrm{P}_{2} } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \mathrm{G}_{2}/\mathrm{P}_{2} } X^5 \ar@{->>}[r]^(.6){\mathbb{P}^{ 2 }} \ar@{->>}[d] & \href{/G2/1 }{ \mathrm{Q}^{5} } \\ \href{/G2/2 }{ \mathrm{G}_{2}/\mathrm{P}_{2} } \ar[r] & X^5 } \end{equation} \begin{equation} \xymatrix{ E_{ \mathrm{Q}^{5} } \ar@{^{(}->}[r] \ar@{->>}[d] & \operatorname{Bl}_{ \mathrm{Q}^{5} } X^5 \ar@{->>}[r]^(.6){\mathbb{P}^{ 2 }} \ar@{->>}[d] & \href{/G2/2 }{ \mathrm{G}_{2}/\mathrm{P}_{2} } \\ \href{/G2/1 }{ \mathrm{Q}^{5} } \ar[r] & X^5 } \end{equation}

The exceptional divisor is the partial flag variety

$ E_{ \mathrm{Q}^{5} } \cong E_{ \mathrm{G}_{2}/\mathrm{P}_{2} } \cong \mathrm{ G }_{ 2 } / \mathrm{P}_{ 1, 2 } $

Exceptional collections

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

Quantum cohomology

small: generically semisimple
big: generically semisimple

Eigenvalues of quantum multiplication by $\mathrm{c}_1(X^5)$:

Homological projective duality

Grassmannian.info

G₂-horospherical variety $X^5$