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We classify special birational transformations of the four-dimensional smoothquadric $Q^4$ where the base locus scheme is assumed to be a smooth surface. Itturns out that there is only one such example, given by the linear system ofcubic complexes of $Q^4$ passing through a non-minimal K3 surface of degree$10$ with $2$ skew $(-1)$-lines.

The cubo-cubic transformation of the smooth quadric fourfold is very special