Construction of the moduli space of reduced Groebner bases

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 Gröbner 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 Gröbner basis scheme, the Gröbner strata and so on if it exists. This paper introduces an alternative procedure for explicitly constructing a defining ideal of the Gröbner 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 Gröbner 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 Gröbner basis scheme for $\prec$ and $J$ defined over any commutative ring $k$ is isomorphic to an affine space over $k$.

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