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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.

MSC:

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

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