Fanography

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

Identification

Fano variety 1-16

hypersurface of degree 2 in $\mathbb{P}^4$

Picard rank
1 (others)
$-\mathrm{K}_X^3$
54
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 1 0
0 0 0 0
0 1 0
0 0
1
1
0 10
0 0 35
0 0 0 30
0 0 0
0 0
0
Anticanonical bundle
index
3
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
30
$-\mathrm{K}_X$ 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

  • 2-7, in a curve of genus 5
  • 2-13, in a curve of genus 2
  • 2-17, in a curve of genus 1
  • 2-21, in a curve of genus 0
  • 2-23, in a curve of genus 1
  • 2-26, in a curve of genus 0
  • 2-29, in a curve of genus 0
  • 2-31, in a curve of genus 0

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

Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PSO}_5$ 10 0
Period sequence

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

GRDB
#3
Fanosearch
#4
Semiorthogonal decompositions

A full exceptional collection was constructed by Kapranov in 1986, see [MR0847146] .

Structure of quantum cohomology

Generic semisimplicity of:

  • small quantum cohomology, proved by someone in at some point, see [?] , using
Zero section description

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

variety
$\mathbb{P}^4$
bundle
$\mathcal{O}(2)$

See the big table for more information.