Gröbner bases and Hilbert schemes. I (Q1117278)

The author studies the family A of all ideals in \(K[x_ 1,...,x_ n]\) that have reduced Gröbner basis with respect to a sequential term ordering \(<_{\sigma}\) on the set T of all monomials in \(x_ 1,...,x_ n\). We obtain that all ideals are of the same dimension and that they allow a parametrization by an affine scheme \(V_ A\) over K. Moreover, if the ordering preserves degrees, then all such ideals have the same Hilbert function. \(V_ A\) is connected; the set of all prime ideals in A, the set of all smooth ideals in A are one to one with the open subsets in \(V_ A.\) For different J in A, Top(J) can have different associated monomial ideals. Finally, since one can find Top of the monomial ideal associated to J, then it is possible to decide whether this ideal is the same as the monomial ideal of Top(J).