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

Mori–Mukai $d$

-4

Still open.

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.

A very general member is not stably rational.

The Weyl group associated to the KKMR decomposition is $\mathrm{A}_1$; see the Weyl-group table for sources and conventions.

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
• MR4719816

For this deformation family, every smooth member has coregularity 0.

The generic and uniform statements are due to Avilov–Loginov–Przyjalkowski; the existence of a smooth member of coregularity zero in every family is due to Zhakupov.

See coregularity for terminology.

For every K-polystable member, the Futaki invariant vanishes identically on the entire Kähler cone.

This is the computation of Sektnan–Tipler.