Fanography

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

Identification

Fano variety 1-14

complete intersection of 2 quadrics in $\mathbb{P}^5$

rank
1 (others)
$-\mathrm{K}_X^3$
32
$\mathrm{h}^{1,2}(X)$
2
Hodge diamond
1
0 0
0 1 0
0 2 2 0
0 1 0
0 0
1
Anticanonical bundle
index
2
del Pezzo of degree 4
$X\hookrightarrow\mathbb{P}^{5}$
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
19
$-\mathrm{K}_X$ 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

  • 2-10, in a curve of genus 1
  • 2-16, in a curve of genus 0
  • 2-19, 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
3

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 3