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 3

0 0 18

0 0 0 22

0 0 0

0 0

0

0 3

0 0 18

0 0 0 22

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 22
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

Birational geometry

Deformation theory

- number of moduli
- 0

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

$\mathrm{PGL}_2$ | 3 | 0 |

Period sequence

Extremal contractions

$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $0\to\mathcal{O}_{\mathbb{P}^2}(-1)^{\oplus2}\to\mathcal{O}_{\mathbb{P}^2}^{\oplus4}\to\mathcal{E}(1)\to 0$.

Semiorthogonal decompositions

A full exceptional collection can be constructed using Orlov's blowup formula.

Structure of quantum cohomology

Generic semisimplicity of:

Zero section description

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

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

See the big table for more information.