Lissité du schéma de Hilbert en bas degré

We check that the Hilbert scheme, , of smooth and connected curves of degree d and genus g in projective three-dimensional space over is smooth provided that d 11. The proof uses essentially our good knowledge of curves lying on cubic surfaces and the possibility to endow a curve having a special normal bundle with a double structure of high arithmetic genus. Then we give some partial results in the case of degree 12. Namely, we obtain that is smooth for g<15 except cases g=11,12, for which we were able to establish only that is smooth in codimension 1. This shows that (12,15) is the lexicographically first pair (d,g) such that is singular in codimension 1.