Fanography

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

Identification

Fano variety 4-1

divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,1,1,1)$

rank
4 (others)
$-\mathrm{K}_X^3$
24
$\mathrm{h}^{1,2}(X)$
1
Hodge diamond
1
0 0
0 4 0
0 1 1 0
0 4 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
15
$-\mathrm{K}_X$ 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 1

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

Deformation theory
number of moduli
3

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 3