A Counterexample to Weitzenb\"ock's Theorem in Characteristic $p$

Abstract

In this paper we give a counterexample to Weitzenb\"{o}ck's Theorem in positive characteristic. Namely we show that if $ k $ is an algebraically closed field of characteristic $ p $, there is an action of $ \mathbb{G}_{a} $ on $ \mathbb{A}^{5}_{k} $, induced by a linear representation of $ \mathbb{G}_{a} $, such that the ring of invariants $ k[x_{1},\dots,x_{5}]^{\mathbb{G}_{a}} $ is not finitely generated.