Fanography

3-29 3-31

Fano variety 3-30

blowup of 2-35 in the proper transform of a line containing the center of the blowup

Alternative description:

• $\mathbb{P}_{\mathbb{F}_1}(\mathcal{O}\oplus\mathcal{O}(\ell))$ where $\ell^2=1$

Picard rank

3 (others)

$-\mathrm{K}_X^3$

50

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

0

index

1

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

28

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 2-33, in a curve of genus 0
• 2-35, in a curve of genus 0

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

• 4-4, in a curve of genus 0
• 4-9, in a curve of genus 0
• 4-12, 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

#11

Fanosearch

#84

$\mathbb{P}^1$-bundle over $\mathbb{F}_1$, for the vector bundle $\pi^*(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(1))$.

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

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

bundle

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

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 #12 on grdb.co.uk.