Fanography

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

Identification

Fano variety 2-9

complete intersection of degree $(1,1)$ and $(2,1)$ in $\mathbb{P}^3\times\mathbb{P}^2$

Alternative description:

  • blowup of 1-17 in a curve of degree 7 and genus 5, which is an intersection of 3 cubics
Picard rank
2 (others)
$-\mathrm{K}_X^3$
16
$\mathrm{h}^{1,2}(X)$
5
Hodge diamond
1
0 0
0 2 0
0 5 5 0
0 2 0
0 0
1
1
0 0
0 13 1
0 0 0 11
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
11
$-\mathrm{K}_X$ 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

  • 1-17, in a curve of genus 5
Deformation theory
number of moduli
13

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 13
Period sequence

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

GRDB
#139
Fanosearch
#62
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
$\mathbb{P}^2 \times \mathbb{P}^3$
bundle
$\mathcal{O}(1,1) \oplus \mathcal{O}(1,2)$

See the big table for more information.