Fanography

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

Identification

Fano variety 1-17

projective space $\mathbb{P}^3$

rank
1 (others)
$-\mathrm{K}_X^3$
64
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 1 0
0 0 0 0
0 1 0
0 0
1
Anticanonical bundle
index
4
del Pezzo of degree 8
$\mathbb{P}^3\hookrightarrow\mathbb{P}^9$, Veronese embedding
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
35
$-\mathrm{K}_X$ 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

  • 2-4, in a curve of genus 10
  • 2-9, in a curve of genus 5
  • 2-12, in a curve of genus 3
  • 2-15, in a curve of genus 4
  • 2-17, in a curve of genus 1
  • 2-19, in a curve of genus 2
  • 2-22, in a curve of genus 0
  • 2-25, in a curve of genus 1
  • 2-27, in a curve of genus 0
  • 2-28, in a curve of genus 1
  • 2-30, in a curve of genus 0
  • 2-33, 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
$\mathrm{PGL}_4$ 15 0