Fanography

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

Identification

Fano variety 5-3

$\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$

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

  • 4-10, in a curve of genus 0

This variety can be blown up (in a curve) to

  • 6-1, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2\times\mathbb{G}_{\mathrm{m}}^2$ 5 0
Period sequence

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

GRDB
#76
Fanosearch
#43
Toric geometry

This variety is toric.

It corresponds to ID #9 on grdb.co.uk.