The scheme of liftings and applications

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 LH by applying the constructive methods of Gröbner 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 LH in a Hilbert scheme as a locally closed subscheme, and, over an infinite field, leads to finding interesting topological properties, as for instance that LH 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.