Hyperelliptic surfaces
A hyperelliptic (or bielliptic) surface is a free quotient of a product of two elliptic curves; equivalently, a smooth projective surface with $\kappa = 0$, $q = 1$, $p_g = 0$. By Bagnera–De Franchis there are exactly seven types, distinguished by the holonomy group (cyclic of order $2$, $3$, $4$ or $6$) and the isogeny type of the torus. All seven share the same Hodge diamond; they differ in the order of the canonical bundle and the number of moduli.
Hodge diamond (shared by all seven)
1
1 1
0 2 0
1 1
1
1 1
0 2 0
1 1
1
| type | holonomy | full group | torus | $\operatorname{ord} \omega_X$ | moduli |
|---|---|---|---|---|---|
| 1 | $\mathrm{C}_{2}$ | $\mathbb{Z}/2$ | $E_\tau \times E_{\tau'}$ | 2 | 2 |
| 2 | $\mathrm{C}_{2}$ | $(\mathbb{Z}/2)^2$ | $(E_\tau \times E_{\tau'})/\langle(\tfrac{\tau}{2},\tfrac12)\rangle$ | 2 | 2 |
| 3 | $\mathrm{C}_{3}$ | $\mathbb{Z}/3$ | $E_\tau \times E_{\zeta_3}$ | 3 | 1 |
| 4 | $\mathrm{C}_{3}$ | $(\mathbb{Z}/3)^2$ | $(E_\tau \times E_{\zeta_3})/\langle(\tfrac{\tau}{3},\tfrac{1-\zeta_3}{3})\rangle$ | 3 | 1 |
| 5 | $\mathrm{C}_{4}$ | $\mathbb{Z}/4$ | $E_\tau \times E_i$ | 4 | 1 |
| 6 | $\mathrm{C}_{4}$ | $\mathbb{Z}/4 \times \mathbb{Z}/2$ | $(E_\tau \times E_i)/\langle(\tfrac{\tau}{4},\tfrac{1-i}{2})\rangle$ | 4 | 1 |
| 7 | $\mathrm{C}_{6}$ | $\mathbb{Z}/6$ | $E_\tau \times E_{\zeta_3}$ | 6 | 1 |
References
- G. Bagnera, M. de Franchis, Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti, Mem. Soc. Ital. Sci. (3) 15 (1908)
- A. Demleitner, Classification of Bagnera–de Franchis varieties in small dimensions, Ann. Fac. Sci. Toulouse Math. (6) 29 (2020) 111–133. MR4809689 doi
- H. Lange, Hyperelliptic varieties, Tôhoku Math. J. (2) 53 (2001) 491–510. MR1862215 doi