  We study the locus of the liftings of a homogeneous ideal $H$ in a polynomial
ring over any field. We prove that this locus can be endowed with a structure
of scheme $\mathrm L_H$ by applying the constructive methods of Gr\"obner
bases, for any given term order. Indeed, this structure does not depend on the
term order, since it can be defined as the scheme representing the functor of
liftings of $H$. We also provide an explicit isomorphism between the schemes
corresponding to two different term orders.
  Our approach allows to embed $\mathrm L_H$ in a Hilbert scheme as a locally
closed subscheme, and, over an infinite field, leads to find interesting
topological properties, as for instance that $\mathrm L_H$ is connected and
that its locus of radical liftings is open. Moreover, we show that every ideal
defining an arithmetically Cohen-Macaulay scheme of codimension two has a
radical lifting, giving in particular an answer to an open question posed by L.
G. Roberts in 1989.
