del Pezzo surface: $\mathrm{Bl}_1\mathbb{P}^2$
Identification
$\mathbb{P}^1\times\mathbb{P}^1$ $\mathrm{Bl}_2\mathbb{P}^2$
del Pezzo surface $\mathrm{Bl}_1\mathbb{P}^2$
- Picard rank
- 2
- $-\mathrm{K}_S^2$
- 8
- alternatives
- divisor of degree $(1,1)$ in $\mathbb{P}^1\times\mathbb{P}^2$
Hodge diamond
1
0 0
0 2 0
0 0
1
0 0
0 2 0
0 0
1
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(S,\omega_S^\vee)$
- 9
- $-\mathrm{K}_S$ very ample?
- yes
Deformation theory
- number of moduli
- 0
- Bott vanishing
- holds
Automorphism groups
| group | dimension | |
| $\mathbb{A}^2\rtimes\mathrm{GL}_2$ | 6 |