On polynomial invariant rings in modular invariant theory

TL;DR

This paper characterizes when the invariant ring under a p-group action is polynomial, confirming a conjecture in specific cases and relating the structure of the invariant ring to the singular locus and Hilbert ideal properties.

Contribution

It proves a characterization of polynomial invariant rings in modular invariant theory and confirms a conjecture for particular cases involving direct summands.

Findings

Invariant ring is polynomial iff singular locus dimension is less than fixed points dimension.

Confirmed conjecture for cases where the invariant ring is a direct summand.

Hilbert ideal is a complete intersection when fixed points are close in rank to the entire space.

Abstract

Let $\Bbbk$ be a field of characteristic $p>0$, $V$ a finite-dimensional $\Bbbk$-vector-space, and $G$ a finite $p$-group acting $\Bbbk$-linearly on $V$. Let $S = \Sym V^*$. We show that $S^G$ is a polynomial ring if and only if the dimension of its singular locus is less than $\rank_\Bbbk V^G$. Confirming a conjecture of Shank-Wehlau-Broer, we show that if $S^G$ is a direct summand of $S$, then $S^G$ is a polynomial ring, in the following cases: \begin{enumerate} \item $\Bbbk = \bbF_p$ and $\rank_\Bbbk V^G = 4$; or \item $|G| = p^3$. \end{enumerate} In order to prove the above result, we also show that if $\rank_\Bbbk V^G \geq \rank_\Bbbk V - 2$, then the Hilbert ideal $\hilbertIdeal_{G,S}$ is a complete intersection.