Identification

Hodge diamond

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

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