Fanography

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

Identification

Fano variety 1-15

section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace

Picard rank
1 (others)
$-\mathrm{K}_X^3$
40
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond and polyvector parallelogram
1
0 0
0 1 0
0 0 0 0
0 1 0
0 0
1
1
0 3
0 0 21
0 0 0 23
0 0 0
0 0
0
Anticanonical bundle
index
2
del Pezzo of degree 5
$X\hookrightarrow\mathbb{P}^{6}$
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
23
$-\mathrm{K}_X$ 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 can be blown up (in a curve) to

  • 2-14, in a curve of genus 1
  • 2-20, in a curve of genus 0
  • 2-22, in a curve of genus 0
  • 2-26, 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
0
Bott vanishing
does not hold
$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2$ 3 0
Period sequence

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

GRDB
#46
Fanosearch
#14
Semiorthogonal decompositions

A full exceptional collection was constructed by Orlov in 1991, see [MR1294662] .

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
$\operatorname{Gr}(2,5)$
bundle
$\mathcal{O}(1)^{\oplus 3}$

See the big table for more information.

Hilbert schemes of curves

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

Its Hodge diamond is

1
0 0
0 1 0
0 0
1