Fanography

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

Identification

Fano variety 1-13

hypersurface of degree 3 in $\mathbb{P}^4$

rank
1 (others)
$-\mathrm{K}_X^3$
24
$\mathrm{h}^{1,2}(X)$
5
Hodge diamond
1
0 0
0 1 0
0 5 5 0
0 1 0
0 0
1
Anticanonical bundle
index
2
del Pezzo of degree 3
$X\hookrightarrow\mathbb{P}^{4}$
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
15
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is not rational but unirational.


This variety is primitive.

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

  • 2-5, in a curve of genus 1
  • 2-11, 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
10

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 10