Fanography

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

del Pezzo surface: $\mathrm{Bl}_7\mathbb{P}^2$

Identification

del Pezzo surface $\mathrm{Bl}_7\mathbb{P}^2$: del Pezzo double plane
Picard rank
8
$-\mathrm{K}_S^2$
2
Hodge diamond
1
0 0
0 8 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(S,\omega_S^\vee)$
3
$-\mathrm{K}_S$ very ample?
no, but $-2\mathrm{K}_S$ is
Deformation theory
number of moduli
6
Automorphism groups
type order structure
I 336 $\mathbb{Z}/2\mathbb{Z}\times\mathrm{PSL}_2(\mathbb{F}_7)$
II 192 $\mathbb{Z}/2\mathbb{Z}\times((\mathbb{Z}/4\mathbb{Z})^2\rtimes\mathrm{Sym}_3)$
III 96 $\mathbb{Z}/2\mathbb{Z}\times(\mathrm{Alt}_4)^4$
IV 48 $\mathbb{Z}/2\mathbb{Z}\times\mathrm{Sym}_4$
V 32 $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}\times(\mathbb{Z}/2\mathbb{Z})^2$
VI 18 $\mathbb{Z}/18\mathbb{Z}$
VII 16 $\mathbb{Z}/2\mathbb{Z}\times\mathrm{Dih}_8$
VIII 12 $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$
IX 12 $\mathbb{Z}/2\mathbb{Z}\times\mathrm{Sym}_3$
X 8 $(\mathbb{Z}/2\mathbb{Z})^3$
XI 6 $\mathbb{Z}/6\mathbb{Z}$
XII 4 $(\mathbb{Z}/2\mathbb{Z})^2$
XIII 2 $\mathbb{Z}/2\mathbb{Z}$