# Fanography

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

Identification
##### Fano variety 2-5

blowup of 1-13 in a plane cubic

rank
2 (others)
$-\mathrm{K}_X^3$
12
$\mathrm{h}^{1,2}(X)$
6
Hodge diamond
1
0 0
0 2 0
0 6 6 0
0 2 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
yes
Birational geometry

This variety is not rational but unirational.

This variety is the blowup of

• 1-13, in a curve of genus 1
Deformation theory
number of moduli
16

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 16