Fanography

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

Identification

Fano variety 1-4

complete intersection of 3 quadrics in $\mathbb{P}^6$

Picard rank
1 (others)
$-\mathrm{K}_X^3$
8
$\mathrm{h}^{1,2}(X)$
14
Hodge diamond
1
0 0
0 1 0
0 14 14 0
0 1 0
0 0
1
1
0 0
0 27 0
0 0 0 7
0 3 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
7
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is not rational but 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
27

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 27
Period sequence

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

GRDB
#163
Fanosearch
#5
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
$\mathbb{P}^6$
bundle
$\mathcal{O}(2)^{\oplus 3}$

See the big table for more information.