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 8
0 0 28
0 0 0 27
0 0 0
0 0
0
0 8
0 0 28
0 0 0 27
0 0 0
0 0
0
Anticanonical bundle
- index
- 2
- del Pezzo of degree 6
- $X\hookrightarrow\mathbb{P}^{7}$
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 27
- $-\mathrm{K}_X$ 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
- 3-7, in a curve of genus 1
- 3-13, in a curve of genus 0
- 3-16, in a curve of genus 0
- 3-20, in a curve of genus 0
- 3-24, in a curve of genus 0
This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.
Deformation theory
- number of moduli
- 0
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$\mathrm{PGL}_3$ | 8 | 0 |
Period sequence
Extremal contractions
$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $\mathrm{T}_{\mathbb{P}^2}$.
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
- $\mathcal{O}(1,1)$
- variety
- $\operatorname{Fl}(1,2,3)$
- bundle
See the big table for more information.
K-stability
- none are K-stable
- every member is K‑polystable
- every member is K‑semistable