Fanography

2-31 2-33

Fano variety 2-32

divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$

Alternative descriptions:

• $\mathbb{P}(\mathrm{T}_{\mathbb{P}^2})$
• the complete flag variety for $\mathbb{P}^2$

Picard rank

2 (others)

$-\mathrm{K}_X^3$

48

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

0

Mori–Mukai $d$

-12

Still open.

index

2

del Pezzo of degree 6

$X\hookrightarrow\mathbb{P}^{7}$

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

27

$-\mathrm{K}_X$ 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 $\mathrm{A}_1$; see the Weyl-group table for sources and conventions.

This variety is primitive.

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

• 3-7, in a curve of genus 1
• 3-13, in a curve of genus 0
• 3-16, in a curve of genus 0
• 3-20, in a curve of genus 0
• 3-24, in a curve of genus 0

This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

number of moduli

0

Bott vanishing

does not hold

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

GRDB

#24

Fanosearch

#6

$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $\mathrm{T}_{\mathbb{P}^2}$.

A full exceptional collection can be constructed using Orlov's projective bundle formula.

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

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

variety

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

bundle

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

variety

$\operatorname{Fl}(1,2,3)$

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.