Fanography

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

Identification

Fano variety 2-12

intersection of 3 $(1,1)$-divisors in $\mathbb{P}^3\times\mathbb{P}^3$

Alternative description:

  • blowup of 1-17 in a curve of degree 6 and genus 3 which is an intersection of 4 cubics
rank
2 (others)
$-\mathrm{K}_X^3$
20
$\mathrm{h}^{1,2}(X)$
3
Hodge diamond
1
0 0
0 2 0
0 3 3 0
0 2 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
13
$-\mathrm{K}_X$ 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

  • 1-17, in a curve of genus 3

This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

Deformation theory
number of moduli
9

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 9