Fanography

2-23 2-25

Fano variety 2-24

divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,2)$

Picard rank

2 (others)

$-\mathrm{K}_X^3$

30

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

0

index

1

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

18

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is primitive.

This variety can be blown up (in a curve) to

• 3-8, in a curve of genus 0

number of moduli

1

Bott vanishing

does not hold

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

GRDB

#44

Fanosearch

#66

$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $0\to\mathcal{O}_{\mathbb{P}^2}(-2)\to\mathcal{O}_{\mathbb{P}^2}^{\oplus3}\to\mathcal{E}(1)\to 0$.

A full exceptional collection can be constructed using Orlov's projective bundle formula.

Generic semisimplicity of:

• small quantum cohomology, proved by Ciolli in 2005, see [MR2168069] , using the description of quantum cohomology of a $\mathbb{P}^1$-bundle

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

variety

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

bundle

$\mathcal{O}(1,2)$

See the big table for more information.

• general member is K‑stable but there exists one that is not
• general member is K‑polystable but there exists one that is not
• general member is K‑semistable

• the book