# Fanography

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

Identification
##### Fano variety 3-4

blowup of 2-18 in a smooth fiber of the composition of the projection to $\mathbb{P}^1\times\mathbb{P}^2$ with the projection to $\mathbb{P}^2$ of the double cover with the projection

rank
3 (others)
$-\mathrm{K}_X^3$
18
$\mathrm{h}^{1,2}(X)$
2
Hodge diamond
1
0 0
0 3 0
0 2 2 0
0 3 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
12
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.

This variety is the blowup of

• 2-18, in a curve of genus 0
Deformation theory
number of moduli
8

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 8