Fanography

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

Identification

Fano variety 1-11

hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$

rank
1 (others)
$-\mathrm{K}_X^3$
8
$\mathrm{h}^{1,2}(X)$
21
Hodge diamond
1
0 0
0 1 0
0 21 21 0
0 1 0
0 0
1
Anticanonical bundle
index
2
del Pezzo of degree 1
$X\to\mathbb{P}^2$
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
7
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is not known to be unirational.


This variety is primitive.

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

  • 2-1, 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
34

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 34