Fanography

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

del Pezzo surface: $\mathrm{Bl}_5\mathbb{P}^2$

Identification

del Pezzo surface $\mathrm{Bl}_5\mathbb{P}^2$: Segre quartic surface
Picard rank
6
$-\mathrm{K}_S^2$
4
Hodge diamond
1
0 0
0 6 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(S,\omega_S^\vee)$
5
$-\mathrm{K}_S$ very ample?
yes
Deformation theory
number of moduli
2
Automorphism groups
type order structure
I 160 $(\mathbb{Z}/2\mathbb{Z})^{\oplus4}\rtimes\mathrm{Dih}_{5}$
II 96 $(\mathbb{Z}/2\mathbb{Z})^{\oplus4}\rtimes\mathrm{Sym}_3$
III 64 $(\mathbb{Z}/2\mathbb{Z})^{\oplus4}\rtimes\mathbb{Z}/4\mathbb{Z}$
IV 32 $(\mathbb{Z}/2\mathbb{Z})^{\oplus4}\rtimes\mathbb{Z}/2\mathbb{Z}$
V 16 $(\mathbb{Z}/2\mathbb{Z})^{\oplus4}$