# Fanography

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

Identification
##### Fano variety 4-7

blowup of 2-32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$

Picard rank
4 (others)
$-\mathrm{K}_X^3$
36
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 4 0
0 0 0 0
0 4 0
0 0
1
1
0 4
0 0 14
0 0 0 21
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
21
$-\mathrm{K}_X$ 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

• 3-24, in a curve of genus 0
• 3-28, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{GL}_2$ 4 0
Period sequence

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

GRDB
#57
Fanosearch
#69
Extremal contractions
Semiorthogonal decompositions

A full exceptional collection can be constructed using Orlov's blowup formula.

Structure of quantum cohomology
Zero section description

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

variety
$(\mathbb{P}^1)^2 \times (\mathbb{P}^2)^2$
bundle
$\mathcal{O}(0,0,1,1) \oplus \mathcal{O}(1,0,1,0) \oplus \mathcal{O}(0,1,0,1)$

variety
$(\mathbb{P}^1)^2 \times \operatorname{Fl}(1,2,3)$
bundle
$\mathcal{O}(1,0; 1,0) \oplus \mathcal{O}(0,1;0,1)$