Fanography

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

Identification

Fano variety 2-34

$\mathbb{P}^1\times\mathbb{P}^2$

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

  • 3-3, in a curve of genus 3
  • 3-5, in a curve of genus 0
  • 3-7, in a curve of genus 1
  • 3-8, in a curve of genus 0
  • 3-11, in a curve of genus 1
  • 3-12, in a curve of genus 0
  • 3-15, in a curve of genus 0
  • 3-17, in a curve of genus 0
  • 3-21, in a curve of genus 0
  • 3-22, in a curve of genus 0
  • 3-24, in a curve of genus 0
  • 3-26, in a curve of genus 0
  • 3-28, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2\times\mathrm{PGL}_3$ 11 0