Fanography

A tool to visually study the geography of Fano 3-folds.

Identification

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
Hodge diamond
1
0 0
0 9 0
0 0 0 0
0 9 0
0 0
1
1
0 3
0 6 3
0 0 18 9
0 0 0
0 0
0
Anticanonical bundle
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
Birational geometry

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
Deformation theory
number of moduli
6

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2$ 3 6
Period sequence

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

Extremal contractions

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

Semiorthogonal decompositions

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

Structure of quantum cohomology

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
Zero section description

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.