Fanography

3-30 4-1

Fano variety 3-31

blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex

Alternative description:

• $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(1,1))$ over $\mathbb{P}^1\times\mathbb{P}^1$

Picard rank

3 (others)

$-\mathrm{K}_X^3$

52

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

0

index

1

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

29

$-\mathrm{K}_X$ 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

• 4-2, in a curve of genus 1
• 4-5, in a curve of genus 0
• 4-8, in a curve of genus 0
• 4-11, in a curve of genus 0
• 4-13, 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

#14

Fanosearch

#53

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

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

$\mathbb{P}^3 \times \mathbb{P}^4$

bundle

$\mathcal{Q}_{\mathbb{P}^3}(0,1) \oplus \mathcal{O}(2,0)$

variety

$\operatorname{Fl}(1,2,5)$

bundle

$\mathcal{Q}_2 \oplus \mathcal{O}(0,2)$

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 #11 on grdb.co.uk.