Seminars: T. Vilkin, Counterexample to the Weitzenböck conjecture

SEMINARS


Seminar of the Laboratory on Algebraic Transformation Groups HSE University February 11, 2026 18:00, Moscow, Pokrovsky b-d 11, D109

Counterexample to the Weitzenböck conjecture T. Vilkin National Research University Higher School of Economics, Moscow
Abstract: Let $G$ be a linear algebraic group. Hilbert’s 14th Problem asks whether, for a linear representation of $G$ on a finite-dimensional vector space over a field $k$, the corresponding invariant ring is finitely generated. It is known to be true for reductive groups. Moreover, for the additive group $\mathrm{G}_a$ over a field of characteristic zero, finite generation was proved by Weitzenböck; a more accessible proof was later given by Seshadri. The question of whether this result remains valid in positive characteristic is known as the Weitzenböck conjecture. In this talk, we will give a counterexample to this conjecture. 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. The talk is based on the paper of S. Maguire «Invariant Rings of $\mathrm{G}_a$-Representations are not always Finitely Generated in Positive Characteristic». arXiv:2509.15431