Identification

Hodge diamond

1

0 0

0 1 0

0 5 5 0

0 1 0

0 0

1

0 0

0 1 0

0 5 5 0

0 1 0

0 0

1

1

0 0

0 10 10

0 0 0 15

0 0 0

0 0

0

0 0

0 10 10

0 0 0 15

0 0 0

0 0

0

Anticanonical bundle

- index
- 2
- del Pezzo of degree 3
- $X\hookrightarrow\mathbb{P}^{4}$
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 15
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

Birational geometry

This variety is not rational but unirational.

This variety is primitive.

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

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

Deformation theory

- number of moduli
- 10

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

$0$ | 0 | 10 |

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
- $\mathbb{P}^4$
- bundle
- $\mathcal{O}(3)$

See the big table for more information.

Hilbert schemes of curves

The **Hilbert scheme of lines** is a Fano surface, a minimal surface of general type.

Its Hodge diamond is

1

5 5

10 25 10

5 5

1

5 5

10 25 10

5 5

1