\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.

G₂-horospherical variety $X^5$

Y Z Y Z
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
$Z=\mathrm{G}_{2}/\mathrm{P}_{2}$ Y Z Y Z unique closed $\mathrm{Aut}(X^5)$-orbit 5 2 3
$Y=\mathrm{Q}^{5}$ Y Z Y Z 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 Y Z Y Z
$ 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 arXiv:1803.05063.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(X^5)$ are given by:

Homological projective duality