del Pezzo surface: $\mathbb{P}^1\times\mathbb{P}^1$
Identification
$\mathbb{P}^2$ $\mathrm{Bl}_1\mathbb{P}^2$
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
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
- Bott vanishing
- holds
Automorphism groups
| group | dimension | |
| $(\mathrm{PGL}_2\times\mathrm{PGL}_2)\rtimes\mathbb{Z}/2\mathbb{Z}$ | 6 |