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 12

0 0 36

0 0 0 31

0 0 0

0 0

0

0 12

0 0 36

0 0 0 31

0 0 0

0 0

0

Anticanonical bundle

- index
- 2
- del Pezzo of degree 7
- $X\hookrightarrow\mathbb{P}^{8}$
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 31
- $-\mathrm{K}_X$ 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

Deformation theory

- number of moduli
- 0
- Bott vanishing
- holds

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

$\mathrm{PGL}_{4;1}$ | 12 | 0 |

Period sequence

Extremal contractions

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

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
- $\mathbb{P}^2 \times \mathbb{P}^3$
- bundle
- $\mathcal{Q}_{\mathbb{P}^2}(0,1)$

- variety
- $\operatorname{Fl}(1,2,4)$
- bundle
- $\mathcal{Q}_2$

See the big table for more information.

K-stability

- none are K-stable
- none are K-polystable
- none are K-semistable

Toric geometry

This variety is toric.

It corresponds to ID #20 on grdb.co.uk.