Fanography

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

Identification

Fano variety 4-10

$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$

rank
4 (others)
$-\mathrm{K}_X^3$
42
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 4 0
0 0 0 0
0 4 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
24
$-\mathrm{K}_X$ 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

  • 3-27, in a curve of genus 0
  • 3-28, in a curve of genus 0

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

  • 5-3, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2\times\mathrm{B}^2$ 7 0