We classify special birational transformations of the four-dimensional smooth quadric $Q^4$ where the base locus scheme is assumed to be a smooth surface. It turns out that there is only one such example, given by the linear system of cubic complexes of $Q^4$ passing through a non-minimal K3 surface of degree $10$ with $2$ skew $(-1)$-lines.PDF: The cubo-cubic transformation of the smooth quadric fourfold is very special.pdf
