Fanography

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

Identification

Fano variety 7-1

$\mathbb{P}^1\times\mathrm{Bl}_5\mathbb{P}^2$

rank
7
$-\mathrm{K}_X^3$
24
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 7 0
0 0 0 0
0 7 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
15
$-\mathrm{K}_X$ 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

  • 6-1, in a curve of genus 0

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

  • 8-1, in a curve of genus 0
Deformation theory
number of moduli
2

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2$ 3 2