Fanography

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

Identification

Fano variety 1-6

section of half-spinor embedding of a connected component of $\mathrm{OGr}_+(5,10)$ by codimension 7 subspace

Picard rank
1 (others)
$-\mathrm{K}_X^3$
12
$\mathrm{h}^{1,2}(X)$
7
Hodge diamond
1
0 0
0 1 0
0 7 7 0
0 1 0
0 0
1
1
0 0
0 18 0
0 0 0 9
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
9
$-\mathrm{K}_X$ 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
18

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 18
Period sequence

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

GRDB
#150
Fanosearch
#7
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
$\operatorname{OGr}^+(5,10)$
bundle
$\mathcal{O}(\frac{1}{2})^{\oplus7}$


variety
$\operatorname{Gr}(2,5)$
bundle
$\mathcal{U}^{\vee}(1)\oplus \mathcal{O}(1)$

See the big table for more information.

Hilbert schemes of curves

The Hilbert scheme of conics is the symmetric square of a smooth curve of genus 7.

Its Hodge diamond is

1
7 7
21 50 21
7 7
1