Quadro-quadric special birational transformations from projective spaces to smooth complete intersections

Abstract: Let \phi: \mathbb{P}^{r}\dashrightarrow Z be a birational transformation with a smooth connected base locus scheme, where Z\subseteq\mathbb{P}^{r+c} is a nondegenerate prime Fano manifold. We call \phi a quadro-quadric special briational transformation if \phi and \phi^{-1} are defined by linear subsystems of |\mathcal{O}{\mathbb{P}^{r}}(2)| and |\mathcal{O}{Z}(2)| respectively. In this paper we classify quadro-quadric special birational transformations in the cases where either (i) Z is a complete intersection and the base locus scheme of \phi^{-1} is smooth, or (ii) Z is a hypersurface.