Fanography

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

Identification

Fano variety 4-4

blowup of 3-19 in the proper transform of a conic through the points

Picard rank
4 (others)
$-\mathrm{K}_X^3$
32
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond and polyvector parallelogram
1
0 0
0 4 0
0 0 0 0
0 4 0
0 0
1
1
0 2
0 0 10
0 0 0 19
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
19
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.

This variety is the blowup of

  • 3-18, in a curve of genus 0
  • 3-19, in a curve of genus 0
  • 3-30, in a curve of genus 0

This variety can be blown up (in a curve) to

  • 5-1, 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
$\mathbb{G}_{\mathrm{m}}^2$ 2 0
Period sequence

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

GRDB
#68
Fanosearch
#103
Extremal contractions
Semiorthogonal decompositions

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

Structure of quantum cohomology
Zero section description

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

variety
$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^4$
bundle
$\mathcal{O}(1,1,0) \oplus \mathcal{O}(0,0,2) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,1)$

See the big table for more information.

K-stability
  • none are K-stable
  • every member is K‑polystable
  • every member is K‑semistable
See and the big table for more information.