Fanography

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

Identification

Fano variety 1-8

section of Plücker embedding of $\mathrm{SGr}(3,6)$ by codimension 3 subspace

Picard rank
1 (others)
$-\mathrm{K}_X^3$
16
$\mathrm{h}^{1,2}(X)$
3
Hodge diamond
1
0 0
0 1 0
0 3 3 0
0 1 0
0 0
1
1
0 0
0 12 0
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 primitive.


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

Deformation theory
number of moduli
12

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 12
Period sequence

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

GRDB
#143
Fanosearch
#20
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
$\operatorname{Gr}(3,6)$
bundle
$\bigwedge^2\mathcal{U}^{\vee} \oplus \mathcal{O}(1)^{\oplus3}$

See the big table for more information.

Hilbert schemes of curves

The Hilbert scheme of conics is the ruled surface obtained from projectivisation of simple rank 2 bundle over a smooth curve of genus 3.

Its Hodge diamond is

1
3 3
0 2 0
3 3
1