|Title||Flat Families by Strongly Stable Ideals and a Generaliza Grobner Bases|
|Author(s)|| Francesca Cioffi, margherita Roggero|
|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 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 Gr ̈bner basis and its constructive capabilities. Indeed, although not every J-marked 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 J-marked|
basis can be characterized by a Buchberger-like criterion. Using J-marked 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 non-standard grading and flat in the origin (the point corresponding to J), thanks to properties of J-marked bases analogous to those of Grobner bases about syzygies.
|Keywords||Family of schemes, strongly stable ideal, Grobner basis, flatness|