Identification

Hodge diamond and polyvector parallelogram

1

0 0

0 2 0

0 0 0 0

0 2 0

0 0

1

0 0

0 2 0

0 0 0 0

0 2 0

0 0

1

1

0 11

0 0 34

0 0 0 30

0 0 0

0 0

0

0 11

0 0 34

0 0 0 30

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 30
- $-\mathrm{K}_X$ 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

- 3-3, in a curve of genus 3
- 3-5, in a curve of genus 0
- 3-7, in a curve of genus 1
- 3-8, in a curve of genus 0
- 3-11, in a curve of genus 1
- 3-12, in a curve of genus 0
- 3-15, in a curve of genus 0
- 3-17, in a curve of genus 0
- 3-21, in a curve of genus 0
- 3-22, in a curve of genus 0
- 3-24, in a curve of genus 0
- 3-26, in a curve of genus 0
- 3-28, in a curve of genus 0

Deformation theory

- number of moduli
- 0
- Bott vanishing
- holds

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

$\mathrm{PGL}_2\times\mathrm{PGL}_3$ | 11 | 0 |

Period sequence

Extremal contractions

$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}$.

Semiorthogonal decompositions

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.

Structure of quantum cohomology

Zero section description

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

- variety
- $\mathbb{P}^1 \times \mathbb{P}^2$
- bundle

See the big table for more information.

K-stability

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

Toric geometry

This variety is toric.

It corresponds to ID #22 on grdb.co.uk.