Fanography

8-1 10-1

Fano variety 9-1

$\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$

Picard rank

9

$-\mathrm{K}_X^3$

12

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

0

index

1

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

9

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

no

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 8-1, in a curve of genus 0

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

• 10-1, in a curve of genus 0

number of moduli

6

Bott vanishing

does not hold

There is no period sequence associated to this Fano 3-fold.

$\mathbb{P}^1$-bundle over $\mathrm{Bl}_7\mathbb{P}^2$, for the vector bundle $\mathcal{O}_{\mathrm{Bl}_7\mathbb{P}^2}\oplus\mathcal{O}_{\mathrm{Bl}_7\mathbb{P}^2}$.

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

$\mathbb{P}(1^3,2) \times \mathbb{P}^1$

bundle

$\mathcal{O}(4,0)$

variety

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

bundle

$\mathcal{O}(2,2,0)$

See the big table for more information.

• none are K-stable
• every member is K‑polystable
• every member is K‑semistable

• the book