2017/07/20 by Kambe, Yuta
#13F20 #13P10 #14C05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1707.06448
For a given monomial ideal J ⊂ k[x1, …, xn] 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 ℚ is nonsingular at the 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.