Fanography

1-1 1-3

Fano variety 1-2

1. hypersurface of degree 4 in $\mathbb{P}^4$
2. double cover of 1-16 with branch locus a divisor of degree 8

Picard rank

1 (others)

$-\mathrm{K}_X^3$

4

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

30

index

1

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

5

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

yes if (a), else no

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

yes

hyperelliptic

no

trigonal

no

This variety is not rational but at least some are 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

1. 45
2. 44

Bott vanishing

does not hold

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

GRDB

#165

Fanosearch

#15

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

$\mathbb{P}^4$

bundle

$\mathcal{O}(4)$

See the big table for more information.

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

• the book