Identification

Hodge diamond

1

0 0

0 2 0

0 11 11 0

0 2 0

0 0

1

0 0

0 2 0

0 11 11 0

0 2 0

0 0

1

1

0 0

0 23 1

0 0 3 7

0 1 0

0 0

0

0 0

0 23 1

0 0 3 7

0 1 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 7
- $-\mathrm{K}_X$ very ample?
- no
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- yes
- trigonal
- no

Birational geometry

This variety is not rational but unirational.

This variety is the blowup of

- 1-12, in a curve of genus 1

Deformation theory

- number of moduli
- 23

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

$0$ | 0 | 23 |

Period sequence

There is no period sequence associated to this Fano 3-fold.

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
- $\mathbb{P}(1^4,2) \times \mathbb{P}^1$
- bundle
- $\mathcal{O}(4,0) \oplus \mathcal{O}(1,1)$

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

See the big table for more information.