Modular invariants of a vector and a covector for some elementary abelian $p$-groups

Let ${\mathbb{F}}_p$ be the prime field of order $p>0$ and $G$ be an elementary abelian $p$-group.For some $n$-dimensional cohyperplane $G$-representations $V$ over ${\mathbb{F}}_p$, we show that ${\mathbb{F}}_p[V\oplus V^{\ast }{]}^G$, the invariant ring of a vector and a covector is a complete intersection by exhibiting an explicit generating set (in fact, a SAGBI basis) and exposing all relations among the generators.