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Let ϕ : ℙr −→ Z a birational transformation with a smooth connected base locus scheme, where Z ⊆ ℙr+c is a nondegenerate prime Fano manifold covered by lines. We call ϕ a quadro-quadric special briational transformation if ϕ and ϕ−1 are defined by linear subsystems of |u0026#x1d4aa;ℙr(2)| and |u0026#x1d4aa;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 ϕ−1 is smooth, or (ii) Z is a hypersurface.

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