Identification

Hodge diamond

1

0 0

0 5 0

0 0 0 0

0 5 0

0 0

1

0 0

0 5 0

0 0 0 0

0 5 0

0 0

1

1

0 5

0 0 13

0 0 0 21

0 0 0

0 0

0

0 5

0 0 13

0 0 0 21

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 21
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

Birational geometry

Deformation theory

- number of moduli
- 0

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

$\mathbb{G}_{\mathrm{m}}\times\mathrm{GL}_2$ | 5 | 0 |

Period sequence

Extremal contractions

Semiorthogonal decompositions

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

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

Structure of quantum cohomology

Generic semisimplicity of:

- small quantum cohomology, proved by Iritani in 2007, see [MR2359850] , using toric geometry

Zero section description

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

- variety
- $\mathbb{P}^1 \times \mathbb{P}^3 \times \mathbb{P}^{8} \times \mathbb{P}^1$
- bundle
- $\mathcal{O}(1,1,0,0) \oplus \Lambda(0,0,1,0) \oplus \mathcal{O}(-1,1,1,0) \oplus \mathcal{O}(0,1,0,1)$

See the big table for more information.

Toric geometry

This variety is toric.

It corresponds to ID #15 on grdb.co.uk.