Fanography

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

Identification

Fano variety 2-35

$\mathrm{Bl}_p\mathbb{P}^3$

Alternative description:

  • $\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(1))$
Picard rank
2 (others)
$-\mathrm{K}_X^3$
56
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond and polyvector parallelogram
1
0 0
0 2 0
0 0 0 0
0 2 0
0 0
1
1
0 12
0 0 36
0 0 0 31
0 0 0
0 0
0
Anticanonical bundle
index
2
del Pezzo of degree 7
$X\hookrightarrow\mathbb{P}^{8}$
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
31
$-\mathrm{K}_X$ 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

  • 3-11, in a curve of genus 1
  • 3-14, in a curve of genus 1
  • 3-16, in a curve of genus 0
  • 3-19, in a curve of genus 0
  • 3-23, in a curve of genus 0
  • 3-26, in a curve of genus 0
  • 3-29, in a curve of genus 0
  • 3-30, in a curve of genus 0
Deformation theory
number of moduli
0
Bott vanishing
holds
$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_{4;1}$ 12 0
Period sequence

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

GRDB
#7
Fanosearch
#30
Extremal contractions

$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(1)$.

Semiorthogonal decompositions

A full exceptional collection can be constructed using Orlov's projective bundle formula.

Alternatively, Kawamata has constructed a full exceptional collection for every smooth projective toric variety.

Structure of quantum cohomology

Generic semisimplicity of:

  • small quantum cohomology, proved by Ciolli in 2005, see [MR2168069] , using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
  • small quantum cohomology, proved by Iritani in 2007, see [MR2359850] , using toric geometry
Zero section description

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

variety
$\mathbb{P}^2 \times \mathbb{P}^3$
bundle
$\mathcal{Q}_{\mathbb{P}^2}(0,1)$


variety
$\operatorname{Fl}(1,2,4)$
bundle
$\mathcal{Q}_2$

See the big table for more information.

Toric geometry

This variety is toric.

It corresponds to ID #20 on grdb.co.uk.