Fanography

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

Identification

Fano variety 2-12

intersection of 3 $(1,1)$-divisors in $\mathbb{P}^3\times\mathbb{P}^3$

Alternative description:

  • blowup of 1-17 in a curve of degree 6 and genus 3 which is an intersection of 4 cubics
Picard rank
2 (others)
$-\mathrm{K}_X^3$
20
$\mathrm{h}^{1,2}(X)$
3
Hodge diamond
1
0 0
0 2 0
0 3 3 0
0 2 0
0 0
1
1
0 0
0 9 3
0 0 0 13
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
13
$-\mathrm{K}_X$ 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 3

This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

Deformation theory
number of moduli
9

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 9
Period sequence

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

GRDB
#118
Fanosearch
#13
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
$\mathbb{P}^3 \times \mathbb{P}^3$
bundle
$\mathcal{O}(1,1)^{\oplus 3}$

See the big table for more information.

Hilbert schemes of curves

The Hilbert scheme of conics is the symmetric square of a genus 3 curve.

Its Hodge diamond is

1
3 3
3 10 3
3 3
1