del Pezzo surface: $\mathrm{Bl}_6\mathbb{P}^2$
Identification
$\mathrm{Bl}_5\mathbb{P}^2$ $\mathrm{Bl}_7\mathbb{P}^2$
del Pezzo surface $\mathrm{Bl}_6\mathbb{P}^2$: cubic surface
- Picard rank
- 7
- $-\mathrm{K}_S^2$
- 3
- alternatives
- triple cover of $\mathbb{P}^2$ branched along a sextic with six cusps lying on a smooth conic
Hodge diamond
1
0 0
0 7 0
0 0
1
0 0
0 7 0
0 0
1
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(S,\omega_S^\vee)$
- 4
- $-\mathrm{K}_S$ very ample?
- yes
Deformation theory
- number of moduli
- 4
- Bott vanishing
- does not hold
Automorphism groups
| type | order | structure |
|---|---|---|
| I | 648 | $3^3:\mathrm{Sym}_4$ |
| II | 120 | $\mathrm{Sym}_5$ |
| III | 108 | $\mathcal{H}_3(3):4$ |
| IV | 54 | $\mathcal{H}_3(3):2$ |
| V | 24 | $\mathrm{Sym}_4$ |
| VI | 12 | $\mathrm{Sym}_3\times 2$ |
| VII | 8 | $8$ |
| VIII | 6 | $\mathrm{Sym}_3$ |
| IX | 4 | $4$ |
| X | 4 | $2^2$ |
| XI | 2 | $2$ |