Fanography

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

Identification

Fano variety 1-1

double cover of $\mathbb{P}^3$ with branch locus a divisor of degree 6

Alternative description:

  • hypersurface of degree 6 in $\mathbb{P}(1,1,1,1,3)$
Picard rank
1 (others)
$-\mathrm{K}_X^3$
2
$\mathrm{h}^{1,2}(X)$
52
Hodge diamond
1
0 0
0 1 0
0 52 52 0
0 1 0
0 0
1
1
0 0
0 68 0
0 0 0 4
0 35 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
4
$-\mathrm{K}_X$ 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 is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

Deformation theory
number of moduli
68

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 68
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
$\mathbb{P}(1^4,3)$
bundle
$\mathcal{O}(6)$


variety
$\mathbb{P}^3 \times \mathbb{P}^{20}$
bundle
$\mathcal{O}(0,2) \oplus K(0,1)$

See the big table for more information.