Fanography

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

del Pezzo surface: $\mathrm{Bl}_6\mathbb{P}^2$

Identification

del Pezzo surface $\mathrm{Bl}_6\mathbb{P}^2$: cubic surface
Picard rank
7
$-\mathrm{K}_S^2$
3
Hodge diamond
1
0 0
0 7 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(S,\omega_S^\vee)$
4
$-\mathrm{K}_S$ very ample?
yes
Deformation theory
number of moduli
4
Automorphism groups
type order structure
I 648 $3^3:\mathrm{Sym}_4$
II 120 $\mathrm{Sym}_5$
III 108 $\mathcal{H}_3(3):4$
IV 54 $\mathcal{H}_3(3):2$
V 24 $\mathrm{Sym}_4$
VI 12 $\mathrm{Sym}_3\times 2$
VII 8 $8$
VIII 6 $\mathrm{Sym}_3$
IX 4 $4$
X 4 $2^2$
XI 2 $2$