Fanography

2-33 2-35

Fano variety 2-34

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

Picard rank

2 (others)

$-\mathrm{K}_X^3$

54

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

0

Mori–Mukai $d$

-9

Still open.

index

1

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

30

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is stably rational.

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

This variety is primitive.

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

• 3-3, in a curve of genus 3
• 3-5, in a curve of genus 0
• 3-7, in a curve of genus 1
• 3-8, in a curve of genus 0
• 3-11, in a curve of genus 1
• 3-12, in a curve of genus 0
• 3-15, in a curve of genus 0
• 3-17, in a curve of genus 0
• 3-21, in a curve of genus 0
• 3-22, in a curve of genus 0
• 3-24, in a curve of genus 0
• 3-26, in a curve of genus 0
• 3-28, in a curve of genus 0

number of moduli

0

Bott vanishing

holds

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

GRDB

#10

Fanosearch

#44

$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}$.

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

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

bundle

See the big table for more information.

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

• the book

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.

This variety is toric.

It corresponds to ID #22 on grdb.co.uk.