Fanography

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

Identification
Fano variety 3-26

blowup of 1-17 in the disjoint union of a point and a line

Alternative description:

• blowup of line on a plane which is section of 2-34 mapping to $\mathbb{P}^2$
Picard rank
3 (others)
$-\mathrm{K}_X^3$
46
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
1
0 8
0 0 25
0 0 0 26
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
26
$-\mathrm{K}_X$ 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

• 2-34, in a curve of genus 0
• 2-35, in a curve of genus 0

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

• 4-9, in a curve of genus 0
Deformation theory
number of moduli
0

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$ 8 0
Period sequence

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

GRDB
#12
Fanosearch
#113
Extremal contractions
Semiorthogonal decompositions

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

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 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
$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^3$
bundle
$\mathcal{O}(1,0,1) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,1)$