Fanography

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

Identification

Fano variety 3-31

blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex

Alternative description:

  • $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(1,1))$ over $\mathbb{P}^1\times\mathbb{P}^1$
rank
3 (others)
$-\mathrm{K}_X^3$
52
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
29
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.


This variety is primitive.

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

  • 4-2, in a curve of genus 1
  • 4-5, in a curve of genus 0
  • 4-8, in a curve of genus 0
  • 4-11, in a curve of genus 0
  • 4-13, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PSO}_{6;1}$ 11 0