# Fanography

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

Identification
##### Fano variety 5-2

blowup of 3-25 in the disjoint union of two exceptional lines on the same irreducible component

Picard rank
5 (others)
$-\mathrm{K}_X^3$
36
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 5 0
0 0 0 0
0 5 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
21
$-\mathrm{K}_X$ 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

• 4-9, in a curve of genus 0
• 4-11, in a curve of genus 0
• 4-12, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathbb{G}_{\mathrm{m}}\times\mathrm{GL}_2$ 5 0
Period sequence

The following period sequences are associated to this Fano 3-fold:

GRDB
#64
Fanosearch
#87
Toric geometry

This variety is toric.

It corresponds to ID #15 on grdb.co.uk.