Fanography

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

Identification

Fano variety 1-10

zero locus of $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ on $\mathrm{Gr}(3,7)$

Picard rank
1 (others)
$-\mathrm{K}_X^3$
22
$\mathrm{h}^{1,2}(X)$
0
Remarks

The unique such threefold where $\mathrm{Aut}(X)\cong\mathrm{PGL}_2$ is called the Mukai–Umemura 3-fold

Hodge diamond
1
0 0
0 1 0
0 0 0 0
0 1 0
0 0
1
1
0 0
0 6 3
0 0 0 14
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
14
$-\mathrm{K}_X$ 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
6

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2$ 3 0
$\mathbb{G}_{\mathrm{a}}$ 1 0
$\mathbb{G}_{\mathrm{m}}$ 1 1
$0$ 0 6
Period sequence

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

GRDB
#113
Fanosearch
#17
Semiorthogonal decompositions

A full exceptional collection was constructed by Kuznetsov in 1996, see [MR1445274] .

Structure of quantum cohomology

Generic semisimplicity of:

  • small quantum cohomology, proved by someone in at some point, see [?] , using
Zero section description

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

variety
$\operatorname{Gr}(3,7)$
bundle
$(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$

See the big table for more information.

Hilbert schemes of curves

The Hilbert scheme of conics is $\mathbb{P}^2$.

Its Hodge diamond is

1
0 0
0 1 0
0 0
1