On Finite Morphisms of Ruled Surfaces

We prove that a surjective morphism Φ:X→X between ruled surfaces is finite and it descends to a finite morphism ϕ:C′→C between base curves of X′ and X. When Φ is restricted to the fibres of X′, it has a constant degree, say a, and then degΦ=a degϕ. In addition, we have several properties on the inverse image of a minimal section and a fibre of X as well as on the direct images. We also investigate precisely the case when both C′ and C are elliptic and X′ is the fibre product C′×CX especially.