Identification

Hodge diamond

1

0 0

0 1 0

0 7 7 0

0 1 0

0 0

1

0 0

0 1 0

0 7 7 0

0 1 0

0 0

1

1

0 0

0 18 0

0 0 0 9

0 0 0

0 0

0

0 0

0 18 0

0 0 0 9

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 9
- $-\mathrm{K}_X$ 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 is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

Deformation theory

- number of moduli
- 18

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

$0$ | 0 | 18 |

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
- $\operatorname{OGr}^+(5,10)$
- bundle
- $\mathcal{O}(\frac{1}{2})^{\oplus7}$

- variety
- $\operatorname{Gr}(2,5)$
- bundle
- $\mathcal{U}^{\vee}(1)\oplus \mathcal{O}(1)$

See the big table for more information.

Hilbert schemes of curves

The **Hilbert scheme of conics** is the symmetric square of a smooth curve of genus 7.

Its Hodge diamond is

1

7 7

21 50 21

7 7

1

7 7

21 50 21

7 7

1