Fanography

3-16 3-18

Fano variety 3-17

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

Picard rank

3 (others)

$-\mathrm{K}_X^3$

36

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

0

index

1

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

21

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 2-34, in a curve of genus 0

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

• 4-3, in a curve of genus 0

number of moduli

0

Bott vanishing

does not hold

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

GRDB

#39

Fanosearch

#37

$\mathbb{P}^1$-bundle over $\mathbb{P}^1\times\mathbb{P}^1$, for the vector bundle $0\to\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(-1,-1)\to\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}^{\oplus3}\to\mathcal{E}(1,1)\to0$.

A full exceptional collection can be constructed using Orlov's blowup 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

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

bundle

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

See the big table for more information.

• none are K-stable
• every member is K‑polystable
• every member is K‑semistable

• the book