Fanography

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

Identification

Fano variety 2-9

complete intersection of degree $(1,1)$ and $(2,1)$ in $\mathbb{P}^3\times\mathbb{P}^2$

Alternative description:

  • blowup of 1-17 in a curve of degree 7 and genus 5, which is intersection of 3 cubics
rank
2 (others)
$-\mathrm{K}_X^3$
16
$\mathrm{h}^{1,2}(X)$
5
Hodge diamond
1
0 0
0 2 0
0 5 5 0
0 2 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
11
$-\mathrm{K}_X$ 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

  • 1-17, in a curve of genus 5
Deformation theory
number of moduli
13

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 13