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 flat families of ideals endowed with a scheme structure whose defining equations can be determined by an algorithmic procedure. A natural flat family that satisfies these requirements is the so-called Gr\"obner stratum, obtained considering ideals sharing the same initial ideal J with respect to a given term ordering. Gr\"obner strata correspond in general to locally closed subschemes of the Hilbert scheme. Some of them turn out to be open, but it has been showed that they are not sufficient to give an open cover of the Hilbert scheme, except few cases. For this reason, several authors developed new algorithmic procedures, based on the combinatorial properties of Borel-fixed ideals, that allow to associate to each ideal J of this type a scheme MfJ, called J-marked scheme, that is in general larger than the Gr\"obner stratum corresponding to the same ideal J. In order to prove that the outcome of these new procedures are consistent with the scheme structure of the Hilbert scheme, in this paper we provide them a solid functorial foundation and show that the algorithmic procedures introduced in previous papers do give the correct equations defining MfJ. We prove that for all the strongly stable ideals J, the marked schemes MfJ can be embedded in a Hilbert scheme as locally closed subschemes, that are open under suitable conditions on J. Marked schemes of this latter type can be very useful for a computational approach to the study of Hilbert schemes. In particular, if we only consider algebras and schemes over a field of null characteristic, marked schemes MfJ with J strongly stable provide, up to the action of the linear group, an open cover of the Hilbert scheme.