Bilinear secants and birational geometry of blowups of ${\mathbb{P}}^n\times {\mathbb{P}}^{n+1}$

We introduce bilinear secant varieties and joins of subvarieties of products of projective spaces, as a generalisation of the classical secant varieties and joins of projective varieties. We show that the bilinear secant varieties of certain rational normal curves of ${\mathbb{P}}^n\times {\mathbb{P}}^{n+1}$ play a central role in the study of the birational geometry of $X_s^{n,n+1}$, its blowup in $s$ points in general position. We show that $X_s^{n,n+1}$ is log Fano, and we compute its effective and movable cones, for $s\le n+2$ and $n\ge 1$ and for $s\le n+3$ and $n\le 2$, and we compute the effective and movable cones of $X_6^{3,4}$.