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

Let $J\subset S=K[x_0,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ be a monomial strongly stable ideal. The collection $\text{Mf}(J)$ of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S\mathrm{/}I$, is called a {\em $J$-marked family}. It can be endowed with a structure of affine scheme, called a {\em $J$-marked scheme}. For special ideals $J$, $J$-marked schemes provide an open cover of the Hilbert scheme $\text{hilbp}$, where $p(t)$ is the Hilbert polynomial of $S\mathrm{/}J$. Those ideals more suitable to this aim are the $m$-truncation ideals ${\underset{‾}{J}}_{\ge m}$ generated by the monomials of degree $\ge m$ in a saturated strongly stable monomial ideal $\underset{‾}{J}$. Exploiting a characterization of the ideals in $\text{Mf}({\underset{‾}{J}}_{\ge m})$ in terms of a Buchberger-like criterion, we compute the equations defining the ${\underset{‾}{J}}_{\ge m}$-marked scheme by a new reduction relation, called {\em superminimal reduction}, and obtain an embedding of $\Mf(\underline{J}{\geq m})$ 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 $\phi_m: \Mf(\underline{J}{\geq m})\hookrightarrow \Mf(\underline{J}_{\geq m+1})$,characterizethose$\phi_m$ that are isomorphisms in terms of the monomial basis of $\underset{‾}{J}$, especially we characterize the minimum integer $m_0$ such that ${\varphi }_m$ is an isomorphism for every $m\ge m_0$.