Fanography

1-2

Fano variety 1-1

double cover of $\mathbb{P}^3$ with branch locus a divisor of degree 6

Alternative description:

• hypersurface of degree 6 in $\mathbb{P}(1,1,1,1,3)$

Picard rank

1 (others)

$-\mathrm{K}_X^3$

2

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

52

index

1

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

4

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

no

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

yes

hyperelliptic

no

trigonal

no

This variety is not known to be 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

68

Bott vanishing

does not hold

There is no period sequence associated to this Fano 3-fold.

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

$\mathbb{P}(1^4,3)$

bundle

$\mathcal{O}(6)$

variety

$\mathbb{P}^3 \times \mathbb{P}^{20}$

bundle

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

See the big table for more information.

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

• the book