Identification

Hodge diamond

1

0 0

0 1 0

0 21 21 0

0 1 0

0 0

1

0 0

0 1 0

0 21 21 0

0 1 0

0 0

1

1

0 0

0 34 3

0 0 0 7

0 7 0

0 0

0

0 0

0 34 3

0 0 0 7

0 7 0

0 0

0

Anticanonical bundle

- index
- 2
- del Pezzo of degree 1
- $X\to\mathbb{P}^2$
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 7
- $-\mathrm{K}_X$ very ample?
- no
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

Birational geometry

This variety is not known to be unirational.

This variety is primitive.

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

- 2-1, 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
- 34

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

$0$ | 0 | 34 |

Period sequence

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

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
- $\mathbb{P}(1^3,2,3)$
- bundle
- $\mathcal{O}(6)$

See the big table for more information.