Fanography

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

Identification

Fano variety 2-3

blowup of 1-12 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system

Picard rank
2 (others)
$-\mathrm{K}_X^3$
8
$\mathrm{h}^{1,2}(X)$
11
Hodge diamond
1
0 0
0 2 0
0 11 11 0
0 2 0
0 0
1
1
0 0
0 23 1
0 0 3 7
0 1 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
7
$-\mathrm{K}_X$ very ample?
no
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
yes
trigonal
no
Birational geometry

This variety is not rational but unirational.


This variety is the blowup of

  • 1-12, in a curve of genus 1
Deformation theory
number of moduli
23

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 23
Period sequence

There is no period sequence associated to this Fano 3-fold.

Extremal contractions
Semiorthogonal decompositions

There exist interesting semiorthogonal decompositions, but this data is not yet added.

Structure of quantum cohomology

By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.

Zero section description

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

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


variety
$\mathbb{P}^3 \times \mathbb{P}^{10} \times \mathbb{P}^1$
bundle
$\Lambda(0,1,0) \oplus \mathcal{O}(0,2,0) \oplus \mathcal{O}(1,0,1)$

See the big table for more information.