Fanography

4-9 4-11

Fano variety 4-10

$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$

Picard rank

4 (others)

$-\mathrm{K}_X^3$

42

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

0

index

1

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

24

$-\mathrm{K}_X$ 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 0
• 3-28, in a curve of genus 0

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

• 5-3, 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

#48

Fanosearch

#142

$\mathbb{P}^1$-bundle over $\mathrm{Bl}_2\mathbb{P}^2$, for the vector bundle $\mathcal{O}_{\mathrm{Bl}_2\mathbb{P}^2}\oplus\mathcal{O}_{\mathrm{Bl}_2\mathbb{P}^2}$.

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

$(\mathbb{P}^1)^3 \times \mathbb{P}^2$

bundle

$\mathcal{O}(1,0,0,1) \oplus \mathcal{O}(0,1,0,1)$

variety

$\mathbb{P}^1\times \mathbb{P}^2 \times \mathbb{P}^3$

bundle

$\mathcal{O}(0,1,1) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,1)$

See the big table for more information.

• none are K-stable
• none are K-polystable
• none are K-semistable

• the book

This variety is toric.

It corresponds to ID #14 on grdb.co.uk.