Details:
Title  Flat families by strongly stable ideals and a generalization of Gröbner bases  Author(s)  Francesca Cioffi, margherita Roggero  Type  Article in Journal  Abstract  Let J be a strongly stable monomial ideal in S = K[x_0, … , x_n] and let Mf(J) be the family of all homogeneous ideals I in S such that the set of all terms outside J is a K vector basis of the quotient S/I. We show that an ideal I belongs to Mf(J) if and only if it is generated by a special set of polynomials, the J marked basis of I , that in some sense generalizes the notion of reduced Gröbner basis and its constructive capabilities. Indeed, although not every J marked basis is a Gröbner basis with respect to some term order, a sort of reduced form modulo I ∈ Mf(J) can be computed for every homogeneous polynomial, so that a J marked basis can be characterized by a Buchbergerlike criterion. Using J marked bases, we prove that the family Mf(J) can be endowed, in a very natural way, with a structure of an affine scheme that turns out to be homogeneous with respect to a nonstandard grading and flat in the origin (the point corresponding to J ), thanks to properties of J marked bases analogous to those of Gröbner bases about syzygies.  Keywords  Family of schemes, Strongly stable ideal, Gröbner basis, Flatness  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717111000800 
Language  English  Journal  Journal of Symbolic Computation  Volume  46  Number  9  Pages  1070  1084  Year  2011  Edition  0  Translation 
No  Refereed 
No 
