Fanography

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

del Pezzo surface: $\mathbb{P}^1\times\mathbb{P}^1$

Identification

del Pezzo surface $\mathbb{P}^1\times\mathbb{P}^1$: quadric surface
Picard rank
2
$-\mathrm{K}_S^2$
8
Hodge diamond
1
0 0
0 2 0
0 0
1
Anticanonical bundle
index
2
$\dim\mathrm{H}^0(S,\omega_S^\vee)$
9
$-\mathrm{K}_S$ very ample?
yes
Deformation theory
number of moduli
0
Automorphism groups
group dimension
$(\mathrm{PGL}_2\times\mathrm{PGL}_2)\rtimes\mathbb{Z}/2\mathbb{Z}$ 6