Identification

Hodge diamond

1

0 0

0 2 0

0 2 2 0

0 2 0

0 0

1

0 0

0 2 0

0 2 2 0

0 2 0

0 0

1

1

0 0

0 6 6

0 0 0 15

0 0 0

0 0

0

0 0

0 6 6

0 0 0 15

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 15
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

Birational geometry

This variety is rational.

This variety is primitive.

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

- 3-4, in a curve of genus 0

Deformation theory

- number of moduli
- 6

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

$0$ | 0 | 6 |

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
- $\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^6$
- bundle
- $\Lambda(0,0,1) \oplus \mathcal{O}(0,0,2)$

See the big table for more information.