Invariant Rings of $ \mathbb{G}_{a} $-Representations are not always Finitely Generated in Positive Characteristic

Abstract

Hilbert's 14th Problem asks the following question. Given a linear representation $ \beta: G \to \operatorname{GL}(\mathbf{V}) $ of a linear algebraic group over a field $ k $ is the ring $ S_{k}(\mathbf{V}^{\ast}) $ a finitely generated $ k $-algebra? For reductive groups the answer is yes. However, in general the answer is no. Nagata provided one of the earliest counterexamples to this claim and his counterexample was extended by Shigeru Mukai. However, if $ G $ is equal to $ \mathbb{G}_{a} $ and the characteristic of $ k $ is equal to zero, then the answer to Hilbert's 14th problem is yes. Roland Weitzenb\"{o}ck first proved this result in 1932 in an article in Acta Mathematica. Seshadri gave a more accessible proof. While Roland Weitzenb\"{o}ck did not conjecture this claim, the question of whether the theorem that bears his name still holds if the characteristic of the base field $ k $ is $ p>0 $ is known as ``the Weitzenb\"{o}ck conjecture''. We aim to use Mukai's strategy to give a counterexample to the Weitzenb\"{o}ck conjecture. Namely, we construct a six dimensional representation over a field of positive characteristic such that the invariant ring is isomorphic to the Cox ring of the blow-up of a toric surface at the identity of the torus. We use the geometry of the underlying toric variety to show that this Cox ring is not finitely generated.