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Composition of points and Mordell-Weil problem for cubic surfaces

D. Kanevsky·Yu. Manin
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Abstract

Let $V$ be a plane smooth cubic curve over a finitely generated field $k.$ The Mordell-Weil theorem for $V$ states that there is a finite subset $P\subset V(k)$ such that the whole $V(k)$ can be obtained from $P$ by 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 if $W(k)$ is infinite.

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D. Kanevsky, Yu. Manin. 2000-11-23. Composition of points and Mordell-Weil problem for cubic surfaces. https://arxiv.org/abs/math/0011198

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