Smoothing of rational \(m\)-ropes (Q2268527)

Let \(C\) be a smooth, irreducible projective curve. A rope \(X\) of multiplicity \(m\geq 2\) over \(C\) is a nowhere reduced scheme \(X\) whose reduced structure is \(C\) and which locally looks like the first infinitesimal neighbourhood of \(C\) inside the total space of a vector bundle of rank \(m-1\). \textit{F. J. Gallego, M. Gonzalez} and \textit{B. P. Purnaprajna} [Compos. Math. 144, No. 3, 673--688 (2008; Zbl 1146.14017)] proved that rational \(3\)-ropes can be smoothed. It is proved that in general rational \(m\)-ropes are smoothable for \(m\geq 3\).