Fanography

2-36 3-2

Fano variety 3-1

double cover of 3-27 with branch locus a divisor of degree $(2,2,2)$

Picard rank

3 (others)

$-\mathrm{K}_X^3$

12

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

8

index

1

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

9

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is not rational but unirational.

This variety is primitive.

This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

number of moduli

17

Bott vanishing

does not hold

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

GRDB

#154

Fanosearch

#22

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

$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^8$

bundle

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

See the big table for more information.

• every member is K‑stable
• every member is K‑polystable
• every member is K‑semistable

• the book