Fanography

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

Identification

Fano variety 1-13

hypersurface of degree 3 in $\mathbb{P}^4$

Picard rank
1 (others)
$-\mathrm{K}_X^3$
24
$\mathrm{h}^{1,2}(X)$
5
Hodge diamond
1
0 0
0 1 0
0 5 5 0
0 1 0
0 0
1
1
0 0
0 10 10
0 0 0 15
0 0 0
0 0
0
Anticanonical bundle
index
2
del Pezzo of degree 3
$X\hookrightarrow\mathbb{P}^{4}$
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
15
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is not rational but unirational.


This variety is primitive.

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

  • 2-5, in a curve of genus 1
  • 2-11, 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
10

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 10
Period sequence

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

GRDB
#106
Fanosearch
#0
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
$\mathbb{P}^4$
bundle
$\mathcal{O}(3)$

See the big table for more information.

Hilbert schemes of curves

The Hilbert scheme of lines is a Fano surface, a minimal surface of general type.

Its Hodge diamond is

1
5 5
10 25 10
5 5
1