Term-ordering free involutive bases

Michela Ceria (1) , Teo Mora (2) , Margherita Roggero (1)

1 Dipartimento di Matematica [Torino]
2 Dipartimento di Matematica [Genova]

In this paper, we consider a monomial ideal $J \triangleleft P:=A[x_1,\dots ,x_n]$, over a commutative ring $A$, and we face the problem of the characterization for the family $\Mf(J)$ of all homogeneous ideals $I \triangleleft P$ such that the $A$-module $P/I$ is free with basis given by the set of terms in the \Gr\ escalier $\cN(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 \cite{Riq1,Riq2, Riq3} and in \cite{J1,J2,J3} and we generalize the construction of $J$-marked bases and term-ordering free reduction process introduced and deeply studied in \cite{BCLR,CR} 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 $\mathcal F(J)$, called stably complete, that allows an explicit description of the family $\Mf(J)$. We obtain stronger results if $J$ is quasi stable, proving that $\mathcal F(J)$ 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.

Domaines: Géométrie algébrique [math.AG]

Michela Ceria, Teo Mora, Margherita Roggero. Term-ordering free involutive bases. MEGA'2013 (Special Issue), Jun 2013, Frankfurt am Main, Allemagne. ⟨hal-01022881⟩