hyperelliptic(.ncag).info

the classification of hyperelliptic varieties in dimensions 2, 3 and 4

Bagnera–De Franchis type 1

The generator acts as a translation of order $2$ on the first factor and as $-1$ on the second, so $E_{\tau'}/\langle -1\rangle \cong \mathbb{P}^1$.

holonomy group
$\mathrm{C}_{2}$, order $2$ · character table
full acting group (with translations)
$\mathbb{Z}/2$
torus
$E_\tau \times E_{\tau'}$
order of $\omega_X$
2
number of moduli
2
irregularity $q = \dim \operatorname{Aut}^0(X)$
1
Albanese
an elliptic curve; the general fiber is an elliptic curve (details)
$\mathbf{D}^{\mathrm{b}}(X)$
indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$
$(1, 2, 2, 2, 1)$
Hodge diamond
1
1 1
0 2 0
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
020
11
1

twisted Hodge numbers $\mathrm{h}^{p,q}(X,\omega_X^{\otimes j})$

Since $\operatorname{ord} \omega_X = 2$, the twists by powers of the canonical bundle form a finite package of $2$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
11
020
11
1
$\omega_X^{\otimes 1}$
0
11
121
11
0

References

  1. 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)
  2. A. Demleitner, Classification of Bagnera–de Franchis varieties in small dimensions, Ann. Fac. Sci. Toulouse Math. (6) 29 (2020) 111–133. MR4809689 doi
  3. H. Lange, Hyperelliptic varieties, Tôhoku Math. J. (2) 53 (2001) 491–510. MR1862215 doi