Fanography

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

Identification

Fano variety 3-30

blowup of 2-35 in the proper transform of a line containing the center of the blowup

Alternative description:

  • $\mathbb{P}_{\mathbb{F}_1}(\mathcal{O}\oplus\mathcal{O}(\ell))$ where $\ell^2=1$
Picard rank
3 (others)
$-\mathrm{K}_X^3$
50
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
1
0 10
0 0 29
0 0 0 28
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
28
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.


This variety is the blowup of

  • 2-33, in a curve of genus 0
  • 2-35, in a curve of genus 0

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

  • 4-4, in a curve of genus 0
  • 4-9, in a curve of genus 0
  • 4-12, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_{4;2,1}$ 10 0
Period sequence

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

GRDB
#11
Fanosearch
#84
Extremal contractions

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

Semiorthogonal decompositions

A full exceptional collection can be constructed using Orlov's blowup 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
$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^3$
bundle
$\mathcal{O}(1,1,0) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,1)$

See the big table for more information.

Toric geometry

This variety is toric.

It corresponds to ID #12 on grdb.co.uk.