# Fanography

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

Identification
##### Fano variety 2-3

blowup of 1-12 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system

rank
2 (others)
$-\mathrm{K}_X^3$
8
$\mathrm{h}^{1,2}(X)$
11
Hodge diamond
1
0 0
0 2 0
0 11 11 0
0 2 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
7
$-\mathrm{K}_X$ very ample?
no
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
yes
trigonal
no
Birational geometry

This variety is not rational but unirational.

This variety is the blowup of

• 1-12, in a curve of genus 1
Deformation theory
number of moduli
23

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 23