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

We give examples over arbitrary fields of rings of invariants that are not finitely generated. The group involved can be as small as three copies of the additive group, as in Mukai's examples over the complex numbers. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell-Weil rank. Our work suggests a generalization of the Morrison-Kawamata cone conjecture from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in dimension 2 in the case of minimal rational elliptic surfaces.