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 4

0 0 14

0 0 0 21

0 0 0

0 0

0

0 4

0 0 14

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
- Bott vanishing
- holds

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

$\mathrm{GL}_2$ | 4 | 0 |

Period sequence

Extremal contractions

Semiorthogonal decompositions

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

Structure of quantum cohomology

Zero section description

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

- variety
- $(\mathbb{P}^1)^2 \times (\mathbb{P}^2)^2$
- bundle
- $\mathcal{O}(0,0,1,1) \oplus \mathcal{O}(1,0,1,0) \oplus \mathcal{O}(0,1,0,1)$

- variety
- $(\mathbb{P}^1)^2 \times \operatorname{Fl}(1,2,3)$
- bundle
- $\mathcal{O}(1,0; 1,0) \oplus \mathcal{O}(0,1;0,1)$

See the big table for more information.

K-stability

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