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 0

0 6 3

0 0 0 14

0 0 0

0 0

0

0 0

0 6 3

0 0 0 14

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 14
- $-\mathrm{K}_X$ 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
- 6

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

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

$\mathbb{G}_{\mathrm{a}}$ | 1 | 0 |

$\mathbb{G}_{\mathrm{m}}$ | 1 | 1 |

$0$ | 0 | 6 |

Period sequence

Semiorthogonal decompositions

A full exceptional collection was constructed by **Kuznetsov** in **1996**, see [MR1445274]
.

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
- $\operatorname{Gr}(3,7)$
- bundle
- $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$

See the big table for more information.

Hilbert schemes of curves

The **Hilbert scheme of conics** 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