Fanography

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

Identification

Fano variety 3-5

blowup of 2-34 in a curve $C$ of degree $(5,2)$ such that $C\hookrightarrow\mathbb{P}^1\times\mathbb{P}^2\to\mathbb{P}^2$ is an embedding

Picard rank
3 (others)
$-\mathrm{K}_X^3$
20
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond and polyvector parallelogram
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
1
0 0*
0 5* 3
0 0 4 13
0 0 0
0 0
0
* indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
13
$-\mathrm{K}_X$ 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

  • 2-34, in a curve of genus 0
Deformation theory
number of moduli
5
Bott vanishing
does not hold
$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathbb{G}_{\mathrm{m}}$ 1 0
$0$ 0 5
Period sequence

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

GRDB
#138
Fanosearch
#109
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
$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^7$
bundle
$\Lambda(0,0,1) \oplus \mathcal{O}(0,1,1)^{\oplus 2}$

See the big table for more information.

K-stability
  • general member is K‑stable but there exists one that is not
  • general member is K‑polystable but there exists one that is not
  • general member is K‑semistable
See and the big table for more information.