Flat families by strongly stable ideals and a generalization of Groebner bases

Let $J$ be a strongly stable monomial ideal in $S=K[x_0,\ldots,x_n]$ and let $\cM(J)$ 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 $\cM(J)$ 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\"obner basis and its constructive capabilities. Indeed, although not every $J$-marked basis is a Gr\"obner basis with respect to some term order, a sort of reduced form modulo $I\in \cM(J)$ 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 $\cM(J)$ 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 Gr\"obner bases about syzygies.