hyperelliptic(.ncag).info

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

Bagnera–De Franchis type 3

The generator translates the first factor by a $3$-torsion point and multiplies the equianharmonic curve $E_{\zeta_3}$ by $\zeta_3$.

holonomy group
$\mathrm{C}_{3}$, order $3$ · character table
full acting group (with translations)
$\mathbb{Z}/3$
torus
$E_\tau \times E_{\zeta_3}$
order of $\omega_X$
3
number of moduli
1
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 1, 0, 0)$
Hodge diamond
1
1 1
0 2 0
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
010
00
0

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

Since $\operatorname{ord} \omega_X = 3$, the twists by powers of the canonical bundle form a finite package of $3$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
11
020
11
1
$\omega_X^{\otimes 1}$
0
01
011
01
0
$\omega_X^{\otimes 2}$
0
10
110
10
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