Fanography

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

Identification

Fano variety 2-31

blowup of 1-16 in a line

Picard rank
2 (others)
$-\mathrm{K}_X^3$
46
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 2 0
0 0 0 0
0 2 0
0 0
1
1
0 7
0 0 26
0 0 0 26
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
26
$-\mathrm{K}_X$ 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

  • 1-16, in a curve of genus 0

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

  • 3-15, in a curve of genus 0
  • 3-20, in a curve of genus 0
  • 3-23, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PSO}_{5;2}$ 7 0
Period sequence

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

GRDB
#15
Fanosearch
#48
Extremal contractions

$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $0\to\mathcal{O}_{\mathbb{P}^2}\to\mathcal{E}\to\mathcal{J}_x\to0$.

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
$\mathbb{P}^2 \times \operatorname{Gr}(2,4)$
bundle
$\mathcal{U}^{\vee}_{\operatorname{Gr}(2,4)}(1,0) \oplus \mathcal{O}(0,1)$

See the big table for more information.