# Fanography

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

Identification

##### Fano variety 10-1

$\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$

rank
10
$-\mathrm{K}_X^3$
6
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 10 0
0 0 0 0
0 10 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
6
$-\mathrm{K}_X$ very ample?
no
$-\mathrm{K}_X$ basepoint free?
no
hyperelliptic
yes
trigonal
no
Birational geometry

This variety is rational.

This variety is the blowup of

• 9-1, in a curve of genus 0
Deformation theory
number of moduli
8

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2$ 3 8