In this paper, we study isomorphism classes of local, Artinian, Gorenstein $k$-algebras $A$ whose maximal ideal $\frak M$ satisfies $\dim _k(\fM ^3/\fM ^4)=1$ by means of Macaulay's inverse system generalizing a recent result by Elias and Rossi. Then we use such results in order to complete the description of the singular locus of the Gorenstein locus of $\Hilb _{11}(\p n)$.

We introduce a generalization of the formal local cohomology module, which we call a formal local cohomology module with respect to a pair of ideals, and study its various properties. We analyze their structure, upper and lower vanishing and non-vanishing properties. There are various exact sequences concerning formal cohomology modules, among them we have a Mayer-Vietoris sequence with respect to pair ideals. Also, we give another proof for a generalized version of the local duality theorems for Gorenstein, Cohen-Macaulay rings, and a generalization of the Grothendieck duality theorem for Gorenstein rings. We discuss the concept of formal grade with respect to a pair of ideals and give some results about this.

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 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 $\MFScheme {J}$, 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 $\MFScheme {J}$ 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 Gr\"obner strata of zero-dimensional ideals, proving that Gr\"obner strata of any ideals are locally closed subschemes of Hilbert schemes.

Let $K$ be a field with rank one valuation and $V$ the valuation domain of $K$. For a subset $E$ of $V$, the ring of integer-valued polynomials on $E$ is \[ \Int (E, V) = \{f \in K[x] \mid f(E) \subseteq V \}. \] A question of interest regarding $\Int (E, V)$ is: for which $E$ is $\Int (E, V)$ a Pr\"{u}fer domain? In this paper, we contribute a partial answer to this question. We classify exactly when $\Int (E, V)$ is Pr\"{u}fer in the case where the elements of $E$ comprise a pseudo-convergent sequence in $V$. Our work expands on earlier results that apply when $V$ is a discrete valuation domain.

Extending the notion of monomial ideals of nested type, we introduce multigraded modules of nested type. We characterize them algebraically, resulting in the explicit construction of their dimension filtration. We compute several important invariants, including their Castelnuovo-Mumford regularity and projective dimension without {us\nobreak {ing}} the construction of their minimal graded free resolution. We show that they are pretty clean and hence sequentially Cohen-Macaulay.