Identification

Hodge diamond

1

0 0

0 1 0

0 0 0 0

0 1 0

0 0

1

0 0

0 1 0

0 0 0 0

0 1 0

0 0

1

1

0 3

0 0 21

0 0 0 23

0 0 0

0 0

0

0 3

0 0 21

0 0 0 23

0 0 0

0 0

0

Anticanonical bundle

- index
- 2
- del Pezzo of degree 5
- $X\hookrightarrow\mathbb{P}^{6}$
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 23
- $-\mathrm{K}_X$ 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

- 2-14, in a curve of genus 1
- 2-20, in a curve of genus 0
- 2-22, in a curve of genus 0
- 2-26, in a curve of genus 0

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

Deformation theory

- number of moduli
- 0

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

$\mathrm{PGL}_2$ | 3 | 0 |

Period sequence

Semiorthogonal decompositions

A full exceptional collection was constructed by **Orlov** in **1991**, see [MR1294662]
.

Structure of quantum cohomology

Generic semisimplicity of:

- small quantum cohomology, proved by someone in at some point, see [?] , using

Zero section description

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

- variety
- $\operatorname{Gr}(2,5)$
- bundle
- $\mathcal{O}(1)^{\oplus 3}$

See the big table for more information.

Hilbert schemes of curves

The **Hilbert scheme of lines** is $\mathbb{P}^2$.

Its Hodge diamond is

1

0 0

0 1 0

0 0

1

0 0

0 1 0

0 0

1