The Hilbert scheme parametrizing subschemes and flat families of subschemes of a given projective space with fixed Hilbert polynomial is a central object in the study of algebraic varieties. In general, it is very difficult to determine explicitly equations defining a Hilbert scheme, since it is constructed as subschemes of a Grassmannian and then of a projective space of very large dimension through the Plücker embedding. In my talk, I will present a new type of flat families, called marked families, that are well suited to compute local and global equations of the Hilbert scheme.

Marked families are a generalization of another type of families called Groebner strata. Consider a polynomial ring with n+1 variable and coefficients in an algebraically closed field of characteristic zero and fix a term ordering sigma. For every monomial ideal I, we define the Groebner stratum of I and sigma as the set of homogeneous ideals having initial ideal I w.r.t. sigma. The marked family of I consists of homogeneous ideals whose quotient algebra is generated as a vector space by the set of monomials not contained in I and, in general, it is larger than every Groebner stratum of I. Groebner strata have a natural structure of affine homogeneous varieties. Their equations can be efficiently computed by using Buchberger’s algorithm to impose that syzygies of the monomial ideal lift to syzygies of every ideal in the stratum. In the case of marked families, lifting syzygies is more difficult because we do not have a term ordering inducing a noetherian polynomial reduction procedure and, therefore, in general there is not an analogue of Buchberger’s criterion. I will focus on the case when the monomial ideal I is strongly stable and I will show how combinatorial properties of strongly stable ideals allow to define a “special” reduction procedure that is again noetherian and that allows to extend the criterion to lift syzygies.

In the second part of the talk, I will present some results obtained using marked families and some ongoing projects related to computation of global equations of the Hilbert scheme, smoothability of zero-dimensional Gorenstein schemes and construction of family of curves with general moduli.