# Fanography

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

Identification
##### Fano variety 2-6: Verra 3-fold
1. $(2,2)$-divisor on $\mathbb{P}^2\times\mathbb{P}^2$
2. double cover of 2-32 with branch locus an anticanonical divisor
Picard rank
2 (others)
$-\mathrm{K}_X^3$
12
$\mathrm{h}^{1,2}(X)$
9
Hodge diamond
1
0 0
0 2 0
0 9 9 0
0 2 0
0 0
1
1
0 0
0 19 0
0 0 0 9
0 1 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
9
$-\mathrm{K}_X$ 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
1. 19
2. 18

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

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

GRDB
#149
Fanosearch
#11
Extremal contractions
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
$\mathbb{P}^2 \times \mathbb{P}^2$
bundle
$\mathcal{O}(2,2)$