Fanography

3-23 3-25

Fano variety 3-24

the fiber product of 2-32 with $\mathrm{Bl}_p\mathbb{P}^2$ over $\mathbb{P}^2$

Alternative description:

• complete intersection of degree $(1,1,0)$ and $(0,1,1)$ in $\mathbb{P}^1\times\mathbb{P}^2\times\mathbb{P}^2$

Picard rank

3 (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

• 2-32, in a curve of genus 0
• 2-34, in a curve of genus 0

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

• 4-7, 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

#31

Fanosearch

#86

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

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

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

bundle

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

See the big table for more information.

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

• the book