Fanography

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

Identification
Fano variety 2-32

divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$

Alternative descriptions:

• $\mathbb{P}(\mathrm{T}_{\mathbb{P}^2})$
• the complete flag variety for $\mathbb{P}^2$
Picard rank
2 (others)
$-\mathrm{K}_X^3$
48
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 2 0
0 0 0 0
0 2 0
0 0
1
1
0 8
0 0 28
0 0 0 27
0 0 0
0 0
0
Anticanonical bundle
index
2
del Pezzo of degree 6
$X\hookrightarrow\mathbb{P}^{7}$
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
27
$-\mathrm{K}_X$ 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

• 3-7, in a curve of genus 1
• 3-13, in a curve of genus 0
• 3-16, in a curve of genus 0
• 3-20, in a curve of genus 0
• 3-24, 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
$\mathrm{PGL}_3$ 8 0
Period sequence

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

GRDB
#24
Fanosearch
#6
Extremal contractions

$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $\mathrm{T}_{\mathbb{P}^2}$.

Semiorthogonal decompositions

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

Structure of quantum cohomology

Generic semisimplicity of:

• small quantum cohomology, proved by Ciolli in 2005, see [MR2168069] , using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
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
$\mathcal{O}(1,1)$

variety
$\operatorname{Fl}(1,2,3)$
bundle