Fanography

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

Identification

Fano variety 1-11

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

Picard 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
1
0 0
0 34 3
0 0 0 7
0 7 0
0 0
0
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?
no
$-\mathrm{K}_X$ 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
Period sequence

There is no period sequence associated to this Fano 3-fold.

Semiorthogonal decompositions

There exist interesting semiorthogonal decompositions, but this data is not yet added.

Structure of quantum cohomology

By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.

Zero section description

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

variety
$\mathbb{P}(1^3,2,3)$
bundle
$\mathcal{O}(6)$

See the big table for more information.