# Fanography

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

Identification
##### Fano variety 1-17

projective space $\mathbb{P}^3$

Picard rank
1 (others)
$-\mathrm{K}_X^3$
64
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 1 0
0 0 0 0
0 1 0
0 0
1
1
0 15
0 0 45
0 0 0 35
0 0 0
0 0
0
Anticanonical bundle
index
4
del Pezzo of degree 8
$\mathbb{P}^3\hookrightarrow\mathbb{P}^9$, Veronese embedding
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
35
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.

This variety is primitive.

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

• 2-4, in a curve of genus 10
• 2-9, in a curve of genus 5
• 2-12, in a curve of genus 3
• 2-15, in a curve of genus 4
• 2-17, in a curve of genus 1
• 2-19, in a curve of genus 2
• 2-22, in a curve of genus 0
• 2-25, in a curve of genus 1
• 2-27, in a curve of genus 0
• 2-28, in a curve of genus 1
• 2-30, in a curve of genus 0
• 2-33, in a curve of genus 0

This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_4$ 15 0
Period sequence

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

GRDB
#1
Fanosearch
#12
Semiorthogonal decompositions

A full exceptional collection was constructed by Beilinson in 1978, see [MR0509388] .

Alternatively, Kawamata has constructed a full exceptional collection for every smooth projective toric variety.

Structure of quantum cohomology

Generic semisimplicity of:

• small quantum cohomology, proved by someone in at some point, see [?] , using
• small quantum cohomology, proved by Iritani in 2007, see [MR2359850] , using toric geometry
Zero section description

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

variety
$\mathbb{P}^3$
bundle