On polynomial invariant rings in modular invariant theory

Let $\mathbb{k}$ be a field of characteristic $p>0$, $V$ a finite-dimensional $\mathbb{k}$-vector-space, and $G$ a finite $p$-group acting $\mathbb{k}$-linearly on $V$. Let $S=\text{Sym}{V}^{\ast }$. We show that $S^G$ is a polynomial ring if and only if the dimension of its singular locus is less than ${\text{rank}}_{\mathbb{k}}{V}^G$. 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 $\mathbb{k}={\text{bbF}}_p$ and ${\text{rank}}_{\mathbb{k}}{V}^G=4$; or \item $\mathrm{\mid }G\mathrm{\mid }=p^3$. \end{enumerate} In order to prove the above result, we also show that if ${\text{rank}}_{\mathbb{k}}{V}^G\ge {\text{rank}}_{\mathbb{k}}V-2$, then the Hilbert ideal ${\text{hilbertIdeal}}_{G,S}$ is a complete intersection.