del Pezzo surface: $\mathrm{Bl}_8\mathbb{P}^2$
Identification
del Pezzo surface $\mathrm{Bl}_8\mathbb{P}^2$
- Picard rank
- 9
- $-\mathrm{K}_S^2$
- 1
- number of exceptional lines
- 240
- alternatives
- sextic surface in $\mathbb{P}(1,1,2,3)$
Hodge diamond and polyvector parallelogram
1
0 0
0 9 0
0 0
1
0 0
0 9 0
0 0
1
1
0 0
0 8 2
0 0
0
0 0
0 8 2
0 0
0
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(S,\omega_S^\vee)$
- 2
- $-\mathrm{K}_S$ very ample?
- no, but $-3\mathrm{K}_S$ is
Deformation theory
- number of moduli
- 8
- Bott vanishing
- does not hold
Automorphism groups
| type | order | structure |
|---|---|---|
| I | 144 | |
| II | 72 | |
| III | 36 | |
| IV | 30 | |
| V | 24 | |
| VI | 24 | |
| VII | 24 | |
| VIII | 20 | |
| IX | 16 | |
| X | 12 | |
| XI | 12 | |
| XII | 12 | |
| XIII | 10 | |
| XIV | 8 | |
| XV | 8 | |
| XVI | 8 | |
| XVII | 6 | |
| XVIII | 6 | |
| XIX | 4 | |
| XX | 4 | |
| XXI | 2 |