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We give examples over arbitrary fields of rings of invariants that are notfinitely generated. The group involved can be as small as three copies of theadditive group, as in Mukai's examples over the complex numbers. The failure offinite generation comes from certain elliptic fibrations or abelian surfacefibrations having positive Mordell-Weil rank.  Our work suggests a generalization of the Morrison-Kawamata cone conjecturefrom Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture indimension 2 in the case of minimal rational elliptic surfaces.

Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves