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In this paper, our purpose is to give a characterization of the generic special cubic fourfold which contains a smooth rational surface of degree 9 not homologous to a complete intersection. As corollaries, we will give an explicit construction of families of smooth surfaces in generic special cubic fourfoldsX  ∈δ{X\in\mathcal{C}_{\delta}}    for6  &lt;  δ  ≤  30{6&lt;\delta\leq 30}    andδ  ≡    0(mod  6   ){\delta\equiv 0~{}(\bmod~{}6)}   . This applies in particular to give an explicit construction of two different liaison class of smooth surfaces in all such special cubic fourfolds with the prescribed invariants.

A note on special cubic fourfolds of small discriminants