Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 2 0
0 0 0 0
0 2 0
0 0
1
0 0
0 2 0
0 0 0 0
0 2 0
0 0
1
1
0 0^{*}
0 1^{*} 10
0 0 0 18
0 0 0
0 0
0
^{*} indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1, 2
0 0^{*}
0 1^{*} 10
0 0 0 18
0 0 0
0 0
0
^{*} indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1, 2
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 18
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
Birational geometry
This variety is rational.
This variety is primitive.
This variety can be blown up (in a curve) to
- 3-8, in a curve of genus 0
Deformation theory
- number of moduli
- 1
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$\mathbb{G}_{\mathrm{m}}^2$ | 2 | 0 |
$\mathbb{G}_{\mathrm{m}}$ | 1 | 0 |
$0$ | 0 | 1 |
Period sequence
Extremal contractions
$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $0\to\mathcal{O}_{\mathbb{P}^2}(-2)\to\mathcal{O}_{\mathbb{P}^2}^{\oplus3}\to\mathcal{E}(1)\to 0$.
Semiorthogonal decompositions
A full exceptional collection can be constructed using Orlov's projective bundle formula.
Structure of quantum cohomology
Generic semisimplicity of:
- small quantum cohomology, proved by Ciolli in 2005, see [MR2168069] , using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
Zero section description
Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):
- variety
- $\mathbb{P}^2 \times \mathbb{P}^2$
- bundle
- $\mathcal{O}(1,2)$
See the big table for more information.
K-stability
- general member is K‑stable but there exists one that is not
- general member is K‑polystable but there exists one that is not
- general member is K‑semistable
See the big table for more information.