Upgraded methods for the effective computation of marked schemes on a strongly stable ideal

Let J ⊂ S = K[x0, . . . , xn] be a monomial strongly stable ideal. The collection Mf(J) of the homogeneous polynomial ideals I, such that the monomials outside J form a K-vector basis of S/I, is called a J-marked family. It can be endowed with a structure of affine scheme, called a J-marked scheme. For special ideals J , J-marked schemes provide an open cover of the Hilbert scheme Hilbnp(t), where p(t) is the Hilbert polynomial of S/J . Those ideals more suitable to this aim are the m-truncation ideals J≥m generated by the monomials of degree ≥ m in a saturated strongly stable monomial ideal J . Exploiting a characterization of the ideals in Mf(J≥m) in terms of a Buchberger-like criterion, we compute the equations defining the J≥m-marked scheme by a new reduction relation, called superminimal reduction, and obtain an embedding of Mf(J≥m) in an affine space of low dimension. In this setting, explicit computations are achievable in many non-trivial cases. Moreover, for every m, we give a closed embedding φm :Mf(J≥m) ↪→Mf(J≥m+1), characterize those φm that are isomorphisms in terms of the monomial basis of J , especially we characterize the minimum integer m0 such that φm is an isomorphism for every m ≥ m0.