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

Let $J \subseteqŠ ‚S=K[x_0,\ldots, x_n]$ be a monomial strongly stable ideal. The collection $\mathcal{M}f(J)$ 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 $\mathcal{H}\textnormal{ilb}_{p(t)}^n$, where $p(t)$ is the Hilbert polynomial of $S/J$. Those ideals more suitable to this aim are the $m$-truncation ideals $\underline{J}_{\geqslant m}$ generated by the monomials of degree $\geqslant m$ in a saturated strongly stable monomial ideal $\underline{J}$. Exploiting a characterization of the ideals in $\mathcal{M}f(\underline{J}_{\geqslant m})$ in terms of a Buchberger-like criterion, we compute the equations defining the $\underline{J}_{\geqslant m}$-marked scheme by a new reduction relation, called superminimal reduction, and obtain an embedding of $\mathcal{M}f(\underline{J}_{\geqslant m})$ 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 $\phi_m: \mathcal{M}f(\underline{J}_{\geqslant m}) \hookrightarrow \mathcal{M}f(\underline{J}_{\geqslant m+1})$, characterize those $\phi_m$ that are isomorphisms in terms of the monomial basis of $\underline{J}$, especially we characterize the minimum integer $m_0$ such that $\phi_m$ is an isomorphism for every $m\geqslant m_0$.