Details:
Title  Flat Families by Strongly Stable Ideals and a Generaliza Grobner Bases  Author(s)  Francesca Cioffi, margherita Roggero  Type  Manual  Abstract  Let J be a strongly stable monomial ideal in S = K[x1 , . . . , xn ] 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 Kvector 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 Jmarked basis of I, that in some sense generalizes the notion of reduced Gr ̈bner basis and its constructive capabilities. Indeed, although not every Jmarked basis is a Grobner basis with respect to some term order, a sort of normal form modulo I ∈ Mf(J) can be computed for every homogeneous polynomial, so that a Jmarked
basis can be characterized by a Buchbergerlike criterion. Using Jmarked bases, we prove that the family Mf(J) can be endowed, in a very natural way, with a structure of 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 Jmarked bases analogous to those of Grobner bases about syzygies.  Keywords  Family of schemes, strongly stable ideal, Grobner basis, flatness  Length  15 
URL 
http://arxiv.org/abs/1101.2866v1 
Language  English  Year  2011  Month  January  Translation 
No  Refereed 
No 
