A linear counterexample to the Fourteenth Problem of Hilbert in dimension eleven

Abstract

Abstract. A family of G a -actions on affine space A m is constructed, each having a non-finitely generated ring of invariants (m ≥ 6). Because these actions are of small degree, they induce linear actions of unipotent groups G n a G a on A 2n+3 for n ≥ 4, and these invariant rings are also non-finitely generated. The smallest such action presented here is for the group G 4 a G a acting linearly on A 11 .

Hilbert’s 14th problem over finite fields and a conjecture on the cone of curves

“…The smallest known representation of an algebraic group for which finite generation fails is Freudenburg's 11-dimensional representation of a unipotent group (G a ) 4 ⋊G a over the rational numbers [8], based on an example by Kuroda [14]. It would be interesting to know whether there are such low-dimensional examples over finite fields.…”