Deformation of finite morphisms and smoothing of ropes

Abstract

In this article we present a unified way to smooth certain multiple structures called ropes on smooth varieties. We prove that most ropes of arbitrary multiplicity, supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely, finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of $1:1$ maps. We apply our general theory to prove the smoothing of ropes of \multiplicity 3 on $\bold P^1$. Even though this article focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension.