  We study an operation, that we call lifting, creating non-isomorphic monomial
curves from a single monomial curve. Our main result says that all but finitely
many liftings of a monomial curve have Cohen-Macaulay tangent cones even if the
tangent cone of the original curve is not Cohen-Macaulay. This implies that the
Betti sequence of the tangent cone is eventually constant under this operation.
Moreover, all liftings have Cohen-Macaulay tangent cones when the original
monomial curve has a Cohen-Macaulay tangent cone. In this case, all the Betti
sequences are nothing but the Betti sequence of the original curve.
