The scheme of liftings and applications

Abstract

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.\ud \ud 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

From grids to pseudo-grids of lines: resolution and seminormality

“…This is essentially due to the fact that τ τ ′ = τ • τ ′ , like it is highlighted in [2]. In case of t-liftings I of a monomial ideal J, J turns out to be the initial ideal of I (for example, see [1,Theorem 3.2]). Thus, we obtain minimal generators of syzygies of I from minimal generators of syzygies of J by a Gröbner basis rewriting procedure, because the Betti numbers coincide.…”

Functors of liftings of projective schemes

“…In this section, referring to Roggero, 2011, 2016) and to (Bertone et al, 2016), we collect some known information about Gröbner functor and functor of x n -liftings. Both these functors are subfunctors of a Hilbert functor, for which we refer to (Grothendieck, 1995, Nitsure, 2005.…”

“…(ii) The family of the liftings of Y with Hilbert polynomial p(t) is parameterized by a subscheme of Hilb n p(t) which can be explicitly constructed. This paper has been motivated by the investigation of x n -liftings of a homogeneous polynomial ideal (see Definition 2.3) that has been faced in (Bertone et al, 2016) from a functorial point of view. However, the question treated here is different and presents new problems to be solved.…”

“…This assumption allows us to exploit the behavior of Gröbner bases with respect to the degree reverse lexicographic order when we need the saturation of homogeneous ideals and of their initial ideals (see Remark 1.2). The study of x n -lifting presented in (Bertone et al, 2016) needed weaker hypotheses on the term order that was indeed chosen in a more general class of term orders (see Definition 4.4.1 in (Kreuzer and Robbiano, 2005), and (Erdös, 1956) for a first generic classification of term orderings).…”

A general framework for Noetherian well ordered polynomial reductions

“…There are many possible applications of the functorial approach to RSs, first of all to the study of Hilbert schemes since the marked schemes Mf J are flat families. In [11] a subfunctor of Mf J for a suitable RS J is used to investigate the set of x n -liftings of a given homogeneous ideal. We conclude with an aplication to the theory of marked bases: for every RS J we can check whether the J -marked sets are bases performing a finite set of reductions.…”