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In this article we give a sufficient condition for a morphism $\varphi$ froma smooth variety $X$ to projective space, finite onto a smooth image, to bedeformed to an embedding. This result puts some theorems on deformation ofmorphisms of curves and surfaces such as $K3$ and general type, obtained by adhoc methods, in a new, more conceptual light. One of the main interests of ourresult is to apply it to the construction of smooth varieties in projectivespace with given invariants. We illustrate this by using our result toconstruct canonically embedded surfaces with $c_1^2=3p_g-7$ and derive someinteresting properties of their moduli spaces. Another interesting applicationof our result is the smoothing of ropes. We obtain a sufficient condition for arope embedded in projective space to be smoothable. As a consequence, we provethat canonically embedded carpets satisfying certain conditions can besmoothed. We also give simple, unified proofs of known theorems on thesmoothing of $1$--dimensional ropes and $K3$ carpets. Our condition fordeforming $\varphi$ to an embedding can be stated very transparently in termsof the cohomology class of a suitable first order infinitesimal deformation of$\varphi$. It holds in a very general setting (any $X$ of arbitrary dimensionand any $\varphi$ unobstructed with an algebraic formally semiuniversaldeformation). The simplicity of the result can be seen for instance when wespecialize it to the case of curves.

An infinitesimal condition to deform a finite morphism to an embedding