Identification

Hodge diamond

1

0 0

0 1 0

0 10 10 0

0 1 0

0 0

1

0 0

0 1 0

0 10 10 0

0 1 0

0 0

1

1

0 0

0 19 6

0 0 0 11

0 1 0

0 0

0

0 0

0 19 6

0 0 0 11

0 1 0

0 0

0

Anticanonical bundle

- index
- 2
- del Pezzo of degree 2
- $X\overset{2:1}{\to}\mathbb{P}^3$
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 11
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

Birational geometry

This variety is not rational but unirational.

This variety is primitive.

This variety can be blown up (in a curve) to

- 2-3, in a curve of genus 1

This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

Deformation theory

- number of moduli
- 19

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

$0$ | 0 | 19 |

Period sequence

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

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

See the big table for more information.