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In this article we show that a wide range of multiple structures on curvesarise whenever a family of embeddings degenerates to a morphism $\varphi$ ofdegree $n$. One could expect to see, when an embedding degenerates to such amorphism, the appearance of a locally Cohen-Macaulay multiple structure ofcertain kind (a so-called rope of multiplicity $n$). We show that thisexpectation is naive and that both locally Cohen-Macaulay and nonCohen-Macaulay multiple structures might occur in this situation. In seeingthis we find out that many multiple structures can be smoothed. When wespecialize to the case of double structures we are able to say much more. Inparticular, we find numerical conditions, in terms of the degree and thearithmetic genus, for the existence of many smoothable double structures. Also,we show that the existence of these double structures is determined, althoughnot uniquely, by the elements of certain space of vector bundle homomorphisms,which are related to the first order infinitesimal deformations of $\varphi$.In many instances, we show that, in order to determine a double structureuniquely, looking merely at a first order deformation of $\varphi$ is notenough; one needs to choose also a formal deformation.

Smoothable locally Cohen--Macaulay and non Cohen--Macaulay multiple structures on curves