# Fanography

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

Identification
##### Fano variety 3-6

blowup of 1-17 in the disjoint union of a line and an elliptic curve of degree 4

Alternative description:

• complete intersection of degree $(1,0,2)$ and $(0,1,1)$ in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^3$
Picard rank
3 (others)
$-\mathrm{K}_X^3$
22
$\mathrm{h}^{1,2}(X)$
1
Hodge diamond
1
0 0
0 3 0
0 1 1 0
0 3 0
0 0
1
1
0 0
0 5 2
0 0 0 14
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
14
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.

This variety is the blowup of

• 2-25, in a curve of genus 0
• 2-33, in a curve of genus 1
Deformation theory
number of moduli
5

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 5
Period sequence

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

GRDB
#117
Fanosearch
#146
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
$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^3$
bundle
$\mathcal{O}(1,0,2) \oplus \mathcal{O}(0,1,1)$

See the big table for more information.