del Pezzo surface: $\mathrm{Bl}_5\mathbb{P}^2$
Identification
$\mathrm{Bl}_4\mathbb{P}^2$ $\mathrm{Bl}_6\mathbb{P}^2$
del Pezzo surface $\mathrm{Bl}_5\mathbb{P}^2$: Segre quartic surface
- Picard rank
- 6
- $-\mathrm{K}_S^2$
- 4
Hodge diamond
1
0 0
0 6 0
0 0
1
0 0
0 6 0
0 0
1
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(S,\omega_S^\vee)$
- 5
- $-\mathrm{K}_S$ very ample?
- yes
Deformation theory
- number of moduli
- 2
- Bott vanishing
- does not hold
Automorphism groups
| type | order | structure |
|---|---|---|
| I | 160 | $(\mathbb{Z}/2\mathbb{Z})^{\oplus4}\rtimes\mathrm{Dih}_{5}$ |
| II | 96 | $(\mathbb{Z}/2\mathbb{Z})^{\oplus4}\rtimes\mathrm{Sym}_3$ |
| III | 64 | $(\mathbb{Z}/2\mathbb{Z})^{\oplus4}\rtimes\mathbb{Z}/4\mathbb{Z}$ |
| IV | 32 | $(\mathbb{Z}/2\mathbb{Z})^{\oplus4}\rtimes\mathbb{Z}/2\mathbb{Z}$ |
| V | 16 | $(\mathbb{Z}/2\mathbb{Z})^{\oplus4}$ |