del Pezzo surface: $\mathrm{Bl}_7\mathbb{P}^2$
Identification
$\mathrm{Bl}_6\mathbb{P}^2$ $\mathrm{Bl}_8\mathbb{P}^2$
del Pezzo surface $\mathrm{Bl}_7\mathbb{P}^2$: del Pezzo double plane
- Picard rank
- 8
- $-\mathrm{K}_S^2$
- 2
- number of exceptional lines
- 56
- alternatives
- quartic surface in $\mathbb{P}(1,1,1,2)$
divisor of degree $(2,2)$ in $\mathbb{P}^1\times\mathbb{P}^2$
Hodge diamond and polyvector parallelogram
1
0 0
0 8 0
0 0
1
0 0
0 8 0
0 0
1
1
0 0
0 6 3
0 0
0
0 0
0 6 3
0 0
0
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(S,\omega_S^\vee)$
- 3
- $-\mathrm{K}_S$ very ample?
- no, but $-2\mathrm{K}_S$ is
Deformation theory
- number of moduli
- 6
- Bott vanishing
- does not hold
Automorphism groups
| type | order | structure |
|---|---|---|
| I | 336 | $\mathbb{Z}/2\mathbb{Z}\times\mathrm{PSL}_2(\mathbb{F}_7)$ |
| II | 192 | $\mathbb{Z}/2\mathbb{Z}\times((\mathbb{Z}/4\mathbb{Z})^2\rtimes\mathrm{Sym}_3)$ |
| III | 96 | $\mathbb{Z}/2\mathbb{Z}\times(\mathrm{Alt}_4)^4$ |
| IV | 48 | $\mathbb{Z}/2\mathbb{Z}\times\mathrm{Sym}_4$ |
| V | 32 | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}\times(\mathbb{Z}/2\mathbb{Z})^2$ |
| VI | 18 | $\mathbb{Z}/18\mathbb{Z}$ |
| VII | 16 | $\mathbb{Z}/2\mathbb{Z}\times\mathrm{Dih}_8$ |
| VIII | 12 | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ |
| IX | 12 | $\mathbb{Z}/2\mathbb{Z}\times\mathrm{Sym}_3$ |
| X | 8 | $(\mathbb{Z}/2\mathbb{Z})^3$ |
| XI | 6 | $\mathbb{Z}/6\mathbb{Z}$ |
| XII | 4 | $(\mathbb{Z}/2\mathbb{Z})^2$ |
| XIII | 2 | $\mathbb{Z}/2\mathbb{Z}$ |