Identification

Hodge diamond

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 0

0 1 7

0 0 0 17

0 0 0

0 0

0

0 0

0 1 7

0 0 0 17

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 17
- $-\mathrm{K}_X$ 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 | 0 |

$0$ | 0 | 1 |

Period sequence

Extremal contractions

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 a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$

Zero section description

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

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

See the big table for more information.