Fanography

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

Identification

Fano variety 3-22

blowup of 2-34 in a conic on $\{x\}\times\mathbb{P}^2$, $x\in\mathbb{P}^1$

rank
3 (others)
$-\mathrm{K}_X^3$
40
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
23
$-\mathrm{K}_X$ 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

  • 2-34, in a curve of genus 0
  • 2-36, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{B}\times\mathrm{PGL}_2$ 5 0