Fanography

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

Identification

Fano variety 2-32

divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$

Alternative descriptions:

  • $\mathbb{P}(\mathrm{T}_{\mathbb{P}^2})$
  • the complete flag variety for $\mathbb{P}^2$
rank
2 (others)
$-\mathrm{K}_X^3$
48
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 2 0
0 0 0 0
0 2 0
0 0
1
Anticanonical bundle
index
2
del Pezzo of degree 6
$X\hookrightarrow\mathbb{P}^{7}$
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
27
$-\mathrm{K}_X$ 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

  • 3-7, in a curve of genus 1
  • 3-13, in a curve of genus 0
  • 3-16, in a curve of genus 0
  • 3-20, in a curve of genus 0
  • 3-24, 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
$\mathrm{PGL}_3$ 8 0