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LetVbe a plane smooth cubic curve over a finitely generated fieldk.The Mordell-Weil theorem for V states that there is a finite subsetP ⠢ V (k)such that the whole V(k) can be obtained fromPby drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with V. Equivalently, the group of birational transformations of V generated by reflections with respect to k-points is finitely generated. In this paper, elaborating an idea from [M3], we establish a Mordell-Weil type finite generation result for some birationally trivial cubic surfaces W. To the contrary, we prove that the birational automorphism group generated by reflections cannot be finitely generated ifW(k)is infinite.

Composition of Points and the Mordell—Weil Problem for Cubic Surfaces