Fanography

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

Identification

Fano variety 2-23
  1. blowup of 1-16 in an intersection of $A\in|\mathcal{O}_Q(1)|$ and $B\in|\mathcal{O}_Q(2)|$ such that $A$ is smooth
  2. blowup of 1-16 in an intersection of $A\in|\mathcal{O}_Q(1)|$ and $B\in|\mathcal{O}_Q(2)|$ such that $A$ is singular
rank
2 (others)
$-\mathrm{K}_X^3$
30
$\mathrm{h}^{1,2}(X)$
1
Hodge diamond
1
0 0
0 2 0
0 1 1 0
0 2 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
18
$-\mathrm{K}_X$ 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

  • 1-16, in a curve of genus 1
Deformation theory
number of moduli
  1. 2
  2. 1

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 2