# Fanography

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

Identification
##### Fano variety 1-5: Gushel–Mukai 3-fold
1. section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 2 subspace and a quadric
2. double cover of 1-15 with branch locus an anticanonical divisor
Picard rank
1 (others)
$-\mathrm{K}_X^3$
10
$\mathrm{h}^{1,2}(X)$
10
Hodge diamond
1
0 0
0 1 0
0 10 10 0
0 1 0
0 0
1
1
0 0
0 22 0
0 0 0 8
0 1 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
8
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is known to be generically non-rational.

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
1. 22
2. 19

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 22
Period sequence

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

GRDB
#160
Fanosearch
#9
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
$\operatorname{Gr}(2,5)$
bundle
$\mathcal{O}(2) \oplus \mathcal{O}(1)^{\oplus2}$