Fanography

2-1 2-3

Fano variety 2-2

double cover of 2-34 with branch locus a $(2,4)$-divisor

Picard rank

2 (others)

$-\mathrm{K}_X^3$

6

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

20

index

1

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

6

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

no

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

yes

hyperelliptic

yes

trigonal

no

This variety is not rational but unirational.

This variety is primitive.

number of moduli

33

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

$\mathbb{P}^1\times \mathbb{P}^2 \times \mathbb{P}^{12}$

bundle

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

See the big table for more information.

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

• the book
• 2210.14770