Identification

Hodge diamond and polyvector parallelogram

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 15

0 0 45

0 0 0 35

0 0 0

0 0

0

0 15

0 0 45

0 0 0 35

0 0 0

0 0

0

Anticanonical bundle

- index
- 4
- del Pezzo of degree 8
- $\mathbb{P}^3\hookrightarrow\mathbb{P}^9$, Veronese embedding
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 35
- $-\mathrm{K}_X$ 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

- 2-4, in a curve of genus 10
- 2-9, in a curve of genus 5
- 2-12, in a curve of genus 3
- 2-15, in a curve of genus 4
- 2-17, in a curve of genus 1
- 2-19, in a curve of genus 2
- 2-22, in a curve of genus 0
- 2-25, in a curve of genus 1
- 2-27, in a curve of genus 0
- 2-28, in a curve of genus 1
- 2-30, in a curve of genus 0
- 2-33, 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
- Bott vanishing
- holds

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

$\mathrm{PGL}_4$ | 15 | 0 |

Period sequence

Semiorthogonal decompositions

A full exceptional collection was constructed by **Beilinson** in **1978**, see [MR0509388]
.

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

Structure of quantum cohomology

Zero section description

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

- variety
- $\mathbb{P}^3$
- bundle

See the big table for more information.

K-stability

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

See the big table for more information.

Toric geometry

This variety is toric.

It corresponds to ID #23 on grdb.co.uk.