Fanography

3-31 4-2

Fano variety 4-1

divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,1,1,1)$

Picard rank

4 (others)

$-\mathrm{K}_X^3$

24

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

1

index

1

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

15

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 3-27, in a curve of genus 1

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

number of moduli

3

Bott vanishing

does not hold

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

GRDB

#111

Fanosearch

#3

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

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

bundle

$\mathcal{O}(1,1,1,1)$

See the big table for more information.

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

• the book
• 2206.12208