Fanography

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

Identification
Fano variety 3-1

double cover of 3-27 with branch locus a divisor of degree $(2,2,2)$

rank
3 (others)
$-\mathrm{K}_X^3$
12
$\mathrm{h}^{1,2}(X)$
8
Hodge diamond
1
0 0
0 3 0
0 8 8 0
0 3 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
9
$-\mathrm{K}_X$ 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 is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

Deformation theory
number of moduli
17

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 17