Smoothing of ribbons over curves

We prove that ribbons, i.e. double structures associated with a line bundle $\text{SE}$ over its reduced support, a smooth irreducible projective curve of arbitrary genus, are smoothable if their arithmetic genus is greater than or equal to $3$ and the support curve possesses a smooth irreducible double cover with trace zero module $\text{SE}$. The method we use is based on the infinitesimal techniques that we develop to show that if the support curve admits such a double cover then every embedded ribbon over the curve is infinitesimally smoothable'', i.e. the ribbon can be obtained as central fiber of the image of some first--order infinitesimal deformation of the map obtained by composing the double cover with the embedding of the reduced support in the ambient projective space containing the ribbon. We also obtain embeddings in the same projective space for all ribbons associated with $\SE$. Then, assuming the existence of the double cover, we prove that the infinitesimal smoothing’’ can be extended to a global embedded smoothing for embedded ribbons of arithmetic genus greater than or equal to 3. As a consequence we obtain the smoothing results.