Upgraded methods for the effective computation of marked schemes on a strongly stable ideal

Details Upgraded methods for the effective computation of marked schemes on a strongly stable ideal Upgraded methods for the effective computation of marked schemes on a strongly stable ideal

2011-10-04

arxiv.org/abs/1110.0698v2

Mathematics

Commutative Algebra

23 pages; this paper contains and extends the second part of the paper posed at arXiv:0909.2184v2[math.AG]; typos corrected in

Scientific paper

Let J be a monomial strongly stable ideal in S=K[x_0,...,x_n]. The collection Mf(J) of the homogeneous polynomial ideals I, such that the monomials outside J form a K-vector basis of S/I, is a family called J-marked family. It can be endowed with a structure of an affine scheme, called J-marked scheme, and then can be embedded as an open subset in the Hilbert scheme Hilb^n_p(t), where p(t) is the Hilbert polynomial of S/J. Exploiting a characterization of the ideals in Mf(J) in terms of a Buchberger-like criterion, we compute a J-marked scheme by a new reduction relation, called superminimal reduction, and obtain an embedding of a J-marked scheme in a linear space of "low" dimension. In this setting, explicit computations are achievable in many non-trivial cases.

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