# Fanography

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

Identification
##### Fano variety 3-2

divisor from $|\mathcal{L}^{\otimes 2}\otimes\mathcal{O}(2,3)|$ on the $\mathbb{P}^2$-bundle $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1,-1)^{\oplus 2})$ over $\mathbb{P}^1\times\mathbb{P}^1$ such that $X\cap Y$ is irreducible, and $\mathcal{L}$ is the tautological bundle, and $Y\in|\mathcal{L}|$

Picard rank
3 (others)
$-\mathrm{K}_X^3$
14
$\mathrm{h}^{1,2}(X)$
3
Hodge diamond
1
0 0
0 3 0
0 3 3 0
0 3 0
0 0
1
1
0 0
0 11 2
0 0 6 10
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
10
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.

This variety is primitive.

Deformation theory
number of moduli
11

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 11
Period sequence

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

GRDB
#157
Fanosearch
#97
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
$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^5$
bundle
$\Lambda(0,0,1) \oplus \mathcal{O}(0,1,2)$

See the big table for more information.