Construction of the moduli space of reduced Groebner bases

TL;DR

This paper develops a new combinatorial method to explicitly construct the moduli space of reduced Gr"obner bases, generalizing previous techniques and providing insights into its geometric structure and singularities.

Contribution

It introduces an alternative, combinatorial approach to construct defining ideals and tangent spaces of Gr"obner basis schemes, extending Robbiano and Lederer's work.

Findings

Provides an explicit construction method for Gr"obner basis schemes.

Shows conditions under which the scheme is isomorphic to an affine space.

Includes an implementation of the proposed method.

Abstract

For a given monomial ideal $J \subset k[x_1, \ldots, x_n]$ and a given monomial order $\prec$, the moduli functor of all reduced Gr\"obner bases with respect to $\prec$ whose initial ideal is $J$ is determined. In some cases, such a functor is representable by an affine scheme of finite type over $k$, and a locally closed subfunctor of a Hilbert scheme. The moduli space is called the Gr\"obner basis scheme, the Gr\"obner strata and so on if it exists. This paper introduces an alternative procedure for explicitly constructing a defining ideal of the Gr\"obner basis scheme and its Zariski tangent spaces by studying combinatorics on the standard set associated to $J$. That is a generalization of Robbiano and Lederer's technique. We also see that we can make an implementation of that. Moreover, as a generalization of Robbiano's result, we show that if the Gr\"obner basis scheme for $\prec$ and $J$ defined over the rational numbers $\mathbb{Q}$ is nonsingular at the $\mathbb{Q}$-rational point corresponding to $J$, then the Gr\"obner basis scheme for $\prec$ and $J$ defined over any commutative ring $k$ is isomorphic to an affine space over $k$.