Term-ordering free involutive bases

Abstract

In this paper, we consider a monomial ideal J ⊳ P := A[x 1 , . . . , x n ], over a commutative ring A, and we face the problem of the characterization for the family Mf (J) of all homogeneous ideals I ⊳ P such that the A-module P/I is free with basis given by the set of terms in the Gröbner escalier N(J) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. any term-ordering, hence more suited to a computational approach to the study of Hilbert schemes. For this purpose, we exploit and enhance the concepts of multiplicative variables, complete sets and involutive bases introduced by Janet in [19,20,21] and we generalize the construction of J-marked bases and term-ordering free reduction process introduced and deeply studied in [1,6] for the special case of a strongly stable monomial ideal J.Here, we introduce and characterize for every monomial ideal J a particular complete set of generators F (J), called stably complete, that allows an explicit description of the family Mf (J). We obtain stronger results if J is quasi stable, proving that F (J) is a Pommaret basis and Mf (J) has a natural structure of affine scheme.The final section presents a detailed analysis of the origin and the historical evolution of the main notions we refer to.

Bar code for monomial ideals

“…The star set F (I) of a monomial ideal I is strongly connected to Janet's theory [27,28,29,30] and to the notion of Pommaret basis [43,44,48], as explicitly pointed out in [12]. For completeness sake, we recall it below.…”

“…Moreover, M is stably complete [48,12] if it is complete and for every τ ∈ M it holds mult M By Proposition 39, the Bar Code gives a simple way to deduce the star set from the Groebner escalier of a zerodimensional monomial ideal.…”

“…With the notation of [31], λ 1 = λ 2 = 2. The partition π uniquely identifies the Bar Code B below: 12 Again, as for stable ideals, we will see that B is admissible and that x π i, j 1 x j−i 2 x i−1 3 belongs to the star set associated to B. x…”

“…To do so, we first introduce a bidimensional structure, called Bar Code which allows, a priori, to represent any (finite 1 ) set of terms M and, if M is an order ideal, to authomatically desume many properties of the corresponding monomial ideal I. For example, a Pommaret basis [48,12] of I can be easily desumed.…”

Functors of liftings of projective schemes

“…The ideal J ≥m is called m-truncation of J and is quasi-stable, because J is. Referring to (Bertone et al, 2017, Ceria et al, 2015, Lella and Roggero, 2016, we now recall the definition of marked functor over the m-truncation of a saturated quasi-stable ideal. The marked functor Mf P(J ≥m ) is a representable subfunctor of the Hilbert functor Hilb n p(t) , where p(t) is the Hilbert polynomial of A[x, x n ]/J, and we denote by Mf P(J ≥m ) its representing scheme (Bertone et al, 2017, Theorem 6.6).…”