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

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

2008-08-05

arxiv.org/abs/0808.0695v1

Mathematics

Algebraic Geometry

26 pages. To appear in Compositio Mathematica

Scientific paper

10.1112/S0010437X08003667

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.

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