Identification

Hodge diamond and polyvector parallelogram

1

0 0

0 4 0

0 0 0 0

0 4 0

0 0

1

0 0

0 4 0

0 0 0 0

0 4 0

0 0

1

1

0 6

0 0 18

0 0 0 23

0 0 0

0 0

0

0 6

0 0 18

0 0 0 23

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 23
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no

Birational geometry

Deformation theory

- number of moduli
- 0
- Bott vanishing
- holds

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

$\mathrm{PGL}_{(2,2);1}$ | 6 | 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
- $(\mathbb{P}^1)^2 \times \mathbb{P}^2 \times \mathbb{P}^3$
- bundle
- $\mathcal{O}(1,0,1,0) \oplus \mathcal{O}(0,1,0,1) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,0,1)$

See the big table for more information.

K-stability

- K-stability is not understood
- none are K-polystable
- none are K-semistable

See the big table for more information.

Toric geometry

This variety is toric.

It corresponds to ID #13 on grdb.co.uk.