Fanography

2-7 2-9

Fano variety 2-8

1. double cover of 2-35 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is smooth
2. double cover of 2-35 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is singular but reduced

Picard rank

2 (others)

$-\mathrm{K}_X^3$

14

$\mathrm{h}^{1,2}(X)$

9

index

1

$\dim\mathrm{H}^0(X,\omega_X^\vee)$

10

$-\mathrm{K}_X$ very ample?

yes

$-\mathrm{K}_X$ basepoint free?

yes

hyperelliptic

no

trigonal

no

This variety is not rational but unirational.

This variety is primitive.

number of moduli

1. 18
2. 17

Bott vanishing

does not hold

The following period sequences are associated to this Fano 3-fold:

GRDB

#144

Fanosearch

#26

There exist interesting semiorthogonal decompositions, but this data is not yet added.

By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.

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

variety

$\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^{12}$

bundle

$\Lambda(0,0,1) \oplus \mathcal{O}(0,0,2)$

See the big table for more information.