Identification

Hodge diamond

1

0 0

0 3 0

0 1 1 0

0 3 0

0 0

1

0 0

0 3 0

0 1 1 0

0 3 0

0 0

1

1

0 1

0 1 12

0 0 0 19

0 0 0

0 0

0

0 1

0 1 12

0 0 0 19

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 19
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

Birational geometry

Deformation theory

- number of moduli
- 1

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

$\mathbb{G}_{\mathrm{m}}$ | 1 | 1 |

Period sequence

Extremal contractions

Semiorthogonal decompositions

*There exist interesting semiorthogonal decompositions, but this data is not yet added.*

Structure of quantum cohomology

By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.

Zero section description

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

- variety
- $\mathbb{P}^3 \times \mathbb{P}^{10} \times \mathbb{P}^2$
- bundle
- $\Lambda(0,1,0) \oplus \mathcal{Q}_{\mathbb{P}^2}(1,0,0) \oplus \mathcal{O}(1,1,0)$

See the big table for more information.