Fanography

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

Identification

Fano variety 4-10

$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$

Picard rank
4 (others)
$-\mathrm{K}_X^3$
42
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 4 0
0 0 0 0
0 4 0
0 0
1
1
0 7
0 0 20
0 0 0 24
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
24
$-\mathrm{K}_X$ 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

  • 3-27, in a curve of genus 0
  • 3-28, in a curve of genus 0

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

  • 5-3, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2\times\mathrm{B}^2$ 7 0
Period sequence

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

GRDB
#48
Fanosearch
#142
Extremal contractions

$\mathbb{P}^1$-bundle over $\mathrm{Bl}_2\mathbb{P}^2$, for the vector bundle $\mathcal{O}_{\mathrm{Bl}_2\mathbb{P}^2}\oplus\mathcal{O}_{\mathrm{Bl}_2\mathbb{P}^2}$.

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


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

See the big table for more information.

Toric geometry

This variety is toric.

It corresponds to ID #14 on grdb.co.uk.