0 0

0 3 0

0 0 0 0

0 3 0

0 0

1

0 9

0 0 27

0 0 0 27

0 0 0

0 0

0

- index
- 2
- del Pezzo of degree 6
- $X\hookrightarrow\mathbb{P}^{7}$
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 27
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

This variety is rational.

This variety is primitive.

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

- 4-1, in a curve of genus 1
- 4-3, in a curve of genus 0
- 4-6, in a curve of genus 0
- 4-8, in a curve of genus 0
- 4-10, in a curve of genus 0
- 4-13, in a curve of genus 0

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

- number of moduli
- 0
- Bott vanishing
- holds

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

$\mathrm{PGL}_2^3$ | 9 | 0 |

$\mathbb{P}^1$-bundle over $\mathbb{P}^1\times\mathbb{P}^1$, for the vector bundle $\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}$.

A full exceptional collection can be constructed using Orlov's projective bundle formula.

Alternatively, Kawamata has constructed a full exceptional collection for every smooth projective toric variety.

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

- variety
- $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$
- bundle

See the big table for more information.

- none are K-stable
- every member is K‑polystable
- every member is K‑semistable

The **Hilbert scheme of conics** is the disjoint union of 6 quadrics.

Its Hodge diamond is

0 0

0 2 0

0 0

1

This variety is toric.

It corresponds to ID #21 on grdb.co.uk.