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Let C ⊂ S := P 1 ×P 1 be an integral curve and ν : X → C its normal- ization. Set L := ν ∗ (OS(1, 0)). Here we give numerical conditions which imply that the pair (X, L) is not the pull-back of a pair (Y, R) with Y a smooth curve of genus q> 0 and R ∈ Pic(Y ).

Coverings of curves and non-existence of certain models of them in the quadric surface