Let X be a smooth curve of genus g. When pi>=3g and d>=pi-2g+1 we show the existence of a double covering gamma:C-->X where C a smooth curve of genus pi with a base-point-free pencil of degree d which is not the pull-back of a pencil on X.
On the existence of primitive pencils for smooth curves
Prym varieties of double coverings of elliptic curves
Stabilité des fibrés $Λ^{p}E_{L}$ et condition de Raynaud
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