Fanography

3-26 3-28

Fano variety 3-27

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

Picard rank

3 (others)

$-\mathrm{K}_X^3$

48

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

0

index

2

del Pezzo of degree 6

$X\hookrightarrow\mathbb{P}^{7}$

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

27

$-\mathrm{K}_X$ 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

• 4-1, in a curve of genus 1
• 4-3, in a curve of genus 0
• 4-6, in a curve of genus 0
• 4-8, in a curve of genus 0
• 4-10, in a curve of genus 0
• 4-13, in a curve of genus 0

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

number of moduli

0

Bott vanishing

holds

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

GRDB

#45

Fanosearch

#21

$\mathbb{P}^1$-bundle over $\mathbb{P}^1\times\mathbb{P}^1$, for the vector bundle $\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}$.

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

Alternatively, Kawamata has constructed a full exceptional collection for every smooth projective toric variety.

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
• small quantum cohomology, proved by Iritani in 2007, see [MR2359850] , using toric geometry

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

variety

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

bundle

See the big table for more information.

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

• the book

The Hilbert scheme of conics is the disjoint union of 6 quadrics.

Its Hodge diamond is

This variety is toric.

It corresponds to ID #21 on grdb.co.uk.