On the Hodge cycle of Prym varieties. (English) Zbl 1112.14032

One considers here Galois coverings \(f:Y\to X\) of degree \(3\) of hyperelliptic curves over \(\mathbb C\). One shows that the hyperelliptic involution lifts to an involution \(\sigma _Y\) of \(Y\). Let \(Z:=Y/\sigma_Y\). One shows that \(\text{Prym}(f)\cong \text{Pic}^0(Z) \times \text{Pic}^0(Z)\). Using this description one constructs a rank \(3\) subgroup of the Néron–Severi group of the Prym variety \(\text{Prym}(f)\). This subgroup generates the algebra of Hodge cycles on \(\text{Prym}(f)\) (for \(X\) general). The last section of the paper deals with certain not Galois degree \(3\) ramified covers.


14H30 Coverings of curves, fundamental group
14H40 Jacobians, Prym varieties
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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