Fanography

2-34 2-36

Fano variety 2-35

$\mathrm{Bl}_p\mathbb{P}^3$

Alternative description:

• $\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(1))$

Picard rank

2 (others)

$-\mathrm{K}_X^3$

56

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

0

index

2

del Pezzo of degree 7

$X\hookrightarrow\mathbb{P}^{8}$

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

31

$-\mathrm{K}_X$ 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

• 3-11, in a curve of genus 1
• 3-14, in a curve of genus 1
• 3-16, in a curve of genus 0
• 3-19, in a curve of genus 0
• 3-23, in a curve of genus 0
• 3-26, in a curve of genus 0
• 3-29, in a curve of genus 0
• 3-30, 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

#7

Fanosearch

#30

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

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

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

bundle

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

variety

$\operatorname{Fl}(1,2,4)$

bundle

$\mathcal{Q}_2$

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 #20 on grdb.co.uk.