Fanography

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

Identification

Fano variety 9-1

$\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$

rank
9
$-\mathrm{K}_X^3$
12
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 9 0
0 0 0 0
0 9 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
9
$-\mathrm{K}_X$ very ample?
no
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.


This variety is the blowup of

  • 8-1, in a curve of genus 0

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

  • 10-1, in a curve of genus 0
Deformation theory
number of moduli
6

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2$ 3 6