Fanography

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

Identification

Fano variety 1-15

section of Pl├╝cker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace

rank
1 (others)
$-\mathrm{K}_X^3$
40
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 1 0
0 0 0 0
0 1 0
0 0
1
Anticanonical bundle
index
2
del Pezzo of degree 5
$X\hookrightarrow\mathbb{P}^{6}$
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
23
$-\mathrm{K}_X$ 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

  • 2-14, in a curve of genus 1
  • 2-20, in a curve of genus 0
  • 2-22, in a curve of genus 0
  • 2-26, 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
$\mathrm{PGL}_2$ 3 0