Identification

Hodge diamond and polyvector parallelogram

1

0 0

0 3 0

0 0 0 0

0 3 0

0 0

1

0 0

0 3 0

0 0 0 0

0 3 0

0 0

1

1

0 6

0 0 21

0 0 0 24

0 0 0

0 0

0

0 6

0 0 21

0 0 0 24

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 24
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

Birational geometry

Deformation theory

- number of moduli
- 0
- Bott vanishing
- holds

$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|

$\mathrm{PGL}_{3;1}$ | 6 | 0 |

Period sequence

Extremal contractions

$\mathbb{P}^1$-bundle over $\mathbb{F}_1$, for the vector bundle $\pi^*(\mathrm{T}_{\mathbb{P}^2})$.

Semiorthogonal decompositions

A full exceptional collection can be constructed using Orlov's blowup 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
- $\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^2$
- bundle
- $\mathcal{O}(1,1,0) \oplus \mathcal{O}(0,1,1)$

See the big table for more information.

K-stability

- none are K-stable
- none are K-polystable
- none are K-semistable