Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals

Let $m,n\ge 3$, $(m-1)(n-1)+2\le p\le mn$, and $u=mn-p$. The set ${\mathbb{R}}^{u\times n\times m}$ of all real tensors with size $u\times n\times m$isonetoonecorrespondingtothesetofbilinearmaps$$\mathbb{R}$^m\times $\mathbb{R}$^n\to $\mathbb{R}$^u$\mathrm{.}Weshowthat$$\mathbb{R}$^{m\times n\times p}$haspluraltypicalranks$p$and$p+1$ if and only if there exists a nonsingular bilinear map ${\mathbb{R}}^m\times {\mathbb{R}}^n\to {\mathbb{R}}^u$. We show that there is a dense open subset $\mathcal{O}$ of $$\mathbb{R}$^{u\times n\times m}$suchthatforany$Y\in\mathscr{O}$, the ideal of maximal minors of a matrix defined by $Y$ in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal. Further, we show that there is a dense open subset $\mathcal{T}$ of ${\mathbb{R}}^{n\times p\times m}$ and continuous surjective open maps $\nu \text{\hspace{0.17em}⁣}:\mathcal{O}\to {\mathbb{R}}^{u\times p}$ and $\sigma \text{\hspace{0.17em}⁣}:\mathcal{T}\to {\mathbb{R}}^{u\times p}$, where $$\mathbb{R}$^{u \times p}$isthesetof$u\times p$matriceswithentriesin$$\mathbb{R}$$, such that if $\nu (Y)=\sigma (T)$, then $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}T=p$ if and only if the ideal of maximal minors of the matrix defined by $Y$ is a real prime ideal.