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

Let m,n≥3m,n≥3, (m−1)(n−1)+2≤p≤mn(m−1)(n−1)+2≤p≤mn, and u=mn−pu=mn−p. The set Ru×n×mRu×n×m of all real tensors with size u×n×mu×n×m is one to one corresponding to the set of bilinear maps Rm×Rn→RuRm×Rn→Ru. We show that Rm×n×pRm×n×p has plural typical ranks p   and p+1p+1 if and only if there exists a nonsingular bilinear map Rm×Rn→RuRm×Rn→Ru. We show that there is a dense open subset OO of Ru×n×mRu×n×m such that for any Y∈OY∈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 TT of Rn×p×mRn×p×m and continuous surjective open maps ν:O→Ru×pν:O→Ru×p and σ:T→Ru×pσ:T→Ru×p, where Ru×pRu×p is the set of u×pu×p matrices with entries in RR, such that if ν(Y)=σ(T)ν(Y)=σ(T), then rankT=p if and only if the ideal of maximal minors of the matrix defined by Y is a real prime ideal.