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 , where p(t) is the Hilbert polynomial of S/J. Those ideals more suitable to this aim are the m-truncation ideals generated by the monomials of degree ⩾m in a saturated strongly stable monomial ideal . Exploiting a characterization of the ideals in in terms of a Buchberger-like criterion, we compute the equations defining the -marked scheme by a new reduction relation, called superminimal reduction, and obtain an embedding of 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 , characterize those ϕm that are isomorphisms in terms of the monomial basis of , especially we characterize the minimum integer m0 such that ϕm is an isomorphism for every m⩾m0.