Identification

Hodge diamond

1

0 0

0 2 0

0 3 3 0

0 2 0

0 0

1

0 0

0 2 0

0 3 3 0

0 2 0

0 0

1

1

0 0

0 9 3

0 0 0 13

0 0 0

0 0

0

0 0

0 9 3

0 0 0 13

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 13
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

Birational geometry

This variety is rational.

This variety is the blowup of

- 1-17, in a curve of genus 3

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

Deformation theory

- number of moduli
- 9

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

$0$ | 0 | 9 |

Period sequence

Extremal contractions

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

See the big table for more information.

Hilbert schemes of curves

The **Hilbert scheme of conics** is the symmetric square of a genus 3 curve.

Its Hodge diamond is

1

3 3

3 10 3

3 3

1

3 3

3 10 3

3 3

1