On the functoriality of marked families

The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define at families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow associating to each ideal $J$ of this type a scheme $\mathbf{Mf}_J$, called a $J$-marked scheme. In this paper, we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that, for all strongly stable ideals $J$, the marked schemes $\mathbf{Mf}_J$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederer's result about Gröbner strata of zero-dimensional ideals, proving that Gröbner strata of any ideals are locally closed subschemes of Hilbert schemes.