Fanography

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

Identification

Fano variety 3-31

blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex

Alternative description:

  • $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(1,1))$ over $\mathbb{P}^1\times\mathbb{P}^1$
Picard rank
3 (others)
$-\mathrm{K}_X^3$
52
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
1
0 11
0 0 31
0 0 0 29
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
29
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.


This variety is primitive.

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

  • 4-2, in a curve of genus 1
  • 4-5, in a curve of genus 0
  • 4-8, in a curve of genus 0
  • 4-11, in a curve of genus 0
  • 4-13, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PSO}_{6;1}$ 11 0
Period sequence

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

GRDB
#14
Fanosearch
#53
Extremal contractions

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

Semiorthogonal decompositions

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.

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
  • small quantum cohomology, proved by Iritani in 2007, see [MR2359850] , using toric geometry
Zero section description

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

variety
$\mathbb{P}^3 \times \mathbb{P}^4$
bundle
$\mathcal{Q}_{\mathbb{P}^3}(0,1) \oplus \mathcal{O}(2,0)$


variety
$\operatorname{Fl}(1,2,5)$
bundle
$\mathcal{Q}_2 \oplus \mathcal{O}(0,2)$

See the big table for more information.

Toric geometry

This variety is toric.

It corresponds to ID #11 on grdb.co.uk.