Fanography

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

Identification

Fano variety 1-12

double cover of $\mathbb{P}^3$ with branch locus a smooth quartic surface

Alternative description:

  • hypersurface of degree 4 in $\mathbb{P}(1,1,1,1,2)$
Picard rank
1 (others)
$-\mathrm{K}_X^3$
16
$\mathrm{h}^{1,2}(X)$
10
Hodge diamond
1
0 0
0 1 0
0 10 10 0
0 1 0
0 0
1
1
0 0
0 19 6
0 0 0 11
0 1 0
0 0
0
Anticanonical bundle
index
2
del Pezzo of degree 2
$X\overset{2:1}{\to}\mathbb{P}^3$
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
11
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is not rational but unirational.


This variety is primitive.

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

  • 2-3, 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
19

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 19
Period sequence

The following period sequences are associated to this Fano 3-fold:

GRDB
#140
Fanosearch
#1
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
$\mathbb{P}(1^4,2)$
bundle
$\mathcal{O}(4)$


variety
$\mathbb{P}^3 \times \mathbb{P}^{10}$
bundle
$\Lambda(0,1) \oplus \mathcal{O}(0,2)$

See the big table for more information.