Construction of the moduli space of reduced Groebner bases

Abstract

For a given monomial ideal $J\subset k[x_1,\dots ,x_n]$ 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 $k$.

Withdraw. There is no new method and new contribution. Please see "Gröbner strata in the {H}ilbert scheme of points", Mathias Lederer, J. Commut. Algebra, Volume 3, Number 3 (2011), 349-404

References in corpus (1)

• Ideals with an assigned initial ideal

Cited by in corpus (1)

• Gröbner scheme in the Hilbert scheme and complete intersection monomial ideals