A note on $p$-bases of a regular affine domain extension

Let $R^p\subseteq R'\subseteq R$ be a tower of commutative rings where $R$ is a regular affine domain over an algebraically closed field of prime characteristic $p$ and $R'$ is a regular domain. Suppose $R$ has a $p$-basis $\{\varphi_1,\dots,\varphi_r\}$ over $R^p$ and $[Q(R')\, :\, Q(R^p)]=p^l$ $(1\leq l\leq r-1)$. For a subset $\Gamma_{r-l}$ of $R$ whose elements satisfy a certain condition on linear independence, let $M_{\Gamma_{r-l}}$ be a set of maximal ideals $\mathfrak{m}$ of $R$ such that $\Gamma_{r-l}$ is a $p$-basis of $R_{\mathfrak{m}}$ over $R'_{\mathfrak{m}'}$ $(\mathfrak{m}'=\mathfrak{m}\cap R')$. We shall characterize this set in a geometrical aspect.