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 1

0 0

0 0 0 20

0 0 0

0 0

0

its dimension takes on the values 1, 2

0 1

^{*}0 0

^{*}140 0 0 20

0 0 0

0 0

0

^{*}indicates jumping of $\operatorname{Aut}^0$its dimension takes on the values 1, 2

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 20
- $-\mathrm{K}_X$ 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 |
---|---|---|

$\mathrm{B}$ | 2 | 0 |

$\mathbb{G}_{\mathrm{m}}$ | 1 | 0 |

Period sequence

Extremal contractions

Semiorthogonal decompositions

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

Structure of quantum cohomology

Generic semisimplicity of:

- small quantum cohomology, proved by Ciolli in 2005, see [MR2168069] , using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$

Zero section description

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

- variety
- $\operatorname{Gr}(2,4) \times \operatorname{Gr}(2,5)$
- bundle
- $\mathcal{Q}_{\operatorname{Gr}(2,4)} \boxtimes \mathcal{U}_{\operatorname{Gr}(2,5)}^{\vee} \oplus \mathcal{O}(1,0) \oplus \mathcal{O}(0,1)^{\oplus 2}$

- variety
- $\operatorname{Fl}(2,3,5)$
- bundle
- $\mathcal{U}_1^{\vee} \oplus \mathcal{O}(1,0) \oplus \mathcal{O}(0,1)^{\oplus 2}$

See the big table for more information.

K-stability

- none are K-stable
- none are K-polystable
- general member is K‑semistable but there exists one that is not